Abstract

When a force is applied to the ocean, fluid parcels are accelerated both locally, by the applied force, and nonlocally, by the pressure gradient forces established to maintain continuity and satisfy the kinematic boundary condition. The net acceleration can be represented through a “rotational force” in the rotational component of the momentum equation. This approach elucidates the correspondence between momentum and vorticity descriptions of the large-scale ocean circulation: if two terms balance pointwise in the rotational momentum equation, then the equivalent two terms balance pointwise in the vorticity equation. The utility of the approach is illustrated for three classical problems: barotropic Rossby waves, wind-driven circulation in a homogeneous basin, and the meridional overturning circulation in an interhemispheric basin. In the hydrostatic limit, it is shown that the rotational forces further decompose into depth-integrated forces that drive the wind-driven gyres and overturning forces that are confined to the basin boundaries and drive the overturning circulation. Potential applications of the approach to diagnosing the output of ocean circulation models, alternative and more accurate formulations of numerical ocean models, the dynamics of boundary layer separation, and eddy forcing of the large-scale ocean circulation are discussed.

1. Introduction

When a force is applied to a Boussinesq fluid, such as the ocean, fluid parcels are accelerated both locally, by the applied force, and nonlocally, by the pressure gradient forces established to maintain nondivergence of the resultant motion and to satisfy the kinematic boundary condition. The net acceleration can be represented through a “rotational force” in the rotational component of the momentum equation.

To illustrate this point, consider the effect of a zonal force that varies with latitude on the circulation in a rectangular basin (Fig. 1). As sketched in Fig. 1, the sum of the applied force and pressure gradient forces is purely rotational, contains both zonal and meridional components, and describes the net acceleration of fluid parcels in response to the applied force.

Fig. 1.

Schematic illustrating the response of a Boussinesq fluid to an applied zonal force F that varies with latitude. The net rotational force Frot is the sum of both the applied force F and the pressure gradient forces pF established to maintain continuity and satisfy the kinematic boundary condition.

Fig. 1.

Schematic illustrating the response of a Boussinesq fluid to an applied zonal force F that varies with latitude. The net rotational force Frot is the sum of both the applied force F and the pressure gradient forces pF established to maintain continuity and satisfy the kinematic boundary condition.

In this paper, we develop a mathematical framework for the rotational momentum balance in both two-dimensional and three-dimensional hydrostatic regimes. We illustrate the approach by diagnosing the relevant rotational forces for a range of classical oceanographic problems: barotropic Rossby waves, barotropic wind-driven circulation, a baroclinic front, and the overturning circulation in a rectangular interhemispheric basin. We choose to focus on established problems for two reasons: (i) to instill confidence in the general approach by diagnosing the net rotational forces in problems that are familiar but also (ii) to identify new dynamical insights into certain aspects of these flows.

There are several benefits to considering the rotational momentum balance:

  • Many of our conceptual models of the ocean circulation are formulated in terms of vorticity due to the absence of pressure gradient terms: see, for example, Sverdrup (1947), Stommel (1948), Stommel and Arons (1960), and standard texts such as Pedlosky (1987) and Vallis (2006). However, it is often simpler to think about force balances than vorticity sources and sinks.

  • The same pointwise balance occurs between equivalent terms in the rotational momentum and vorticity equations. If two terms balance pointwise in the vorticity equation, for example, the wind stress curl and advection of planetary vorticity, then the equivalent two terms, the wind stress and Coriolis force, balance pointwise in the rotational momentum equation.

  • For two-dimensional flow, the rotational forces can be defined and plotted in terms of scalar “force functions,” analogous to the streamfunction used to represent two-dimensional velocity fields.

  • In the hydrostatic limit, a three-dimensional rotational force is readily decomposed into a depth-integrated force, which is described by a horizontal force function, and overturning forces, which are described by two overturning force functions. These scalar force functions represent powerful diagnostics for visualizing the three-dimensional momentum balance in ocean models.

  • The nonlocality of the response to an applied force is made explicit, highlighting the pitfall of statements such as “in the Northern Hemisphere the flow is deflected to the right by the Coriolis force,” which ignores that Coriolis forces are mostly compensated by pressure gradient forces (“geostrophic balance”), the residual force being small and oriented in any direction.

  • The approach has a number of immediate applications: for example, reducing errors due to the representation of geostrophic and hydrostatic balance in unstructured-mesh ocean circulation models (Maddison et al. 2011, manuscript submitted to Ocean Modell.).

The paper is structured as follows: In section 2, we present the theoretical background, defining the Helmholtz decomposition of forces into their rotational and divergent components. In section 3, we apply the decomposition to the phase and group propagation of barotropic Rossby waves. In section 4, we diagnose the rotational forces maintaining the wind-driven circulation in a homogeneous basin. In section 5, we discuss the rotational forces acting on a baroclinic front in thermal wind balance. In section 6, we show that the rotational forces can be further decomposed in the hydrostatic limit into depth-integrated and overturning components. In section 7, we diagnose the rotational forces driving the meridional overturning circulation (MOC) in a closed interhemispheric basin. Finally, in section 8, we briefly summarize our main results before discussing potential applications of the new approach.

2. Theoretical background

a. Helmholtz decomposition of a force

Consider an arbitrary force F acting on an incompressible (or Boussinesq) fluid. The momentum equation is

 
formula

where

 
formula

u is the fluid velocity, p is pressure, ρ0 is a reference density, and t is time.

Helmholtz’s theorem states that any vector can be decomposed into purely rotational and divergent components:

 
formula

where

 
formula

The rotational component of the force projects purely onto the local acceleration ∂u/∂t, which is itself purely rotational. To satisfy the kinematic boundary condition, we must therefore impose a condition of no rotational forcing through the solid boundary,

 
formula

where n is a unit vector normal to the boundary. The application of this boundary condition introduces uniqueness in the solution for Frot.

In contrast, the divergent component of the force projects entirely onto the pressure gradient,

 
formula

where pF is the component of pressure associated with F. The net acceleration due to F is thus

 
formula

As discussed above, the net acceleration due to F is the sum of the local applied force and the nonlocal pressure gradient forces established to maintain incompressibility.

We now rewrite the momentum equation by applying this decomposition to each of the forces in turn,

 
formula

Taking the curl of (8) gives the vorticity equation

 
formula

Because there are no pressure gradients in the rotational momentum equation, any pointwise balance existing between two terms in the vorticity equation in (9) will also exist between the associated accelerations in the rotational momentum equation in (8).

b. Definition of force function in two-dimensional flows

In sections 3 and 4, we consider two-dimensional flows for which the rotational forces are conveniently described by a scalar potential field. We call this field the force function, analogous to the use of a streamfunction to describe two-dimensional incompressible flow. Thus, the force function, φF, is defined by

 
formula

where φF = constant along any boundaries to satisfy the boundary condition (5). The absolute value of φF is arbitrary, but in a closed basin it is convenient to set φF = 0 on the boundaries. The rotational force is directed along contours of φF with a strength inversely proportional to the spacing of the contours.

Taking the curl and divergence of (3) yields two Poisson equations:

 
formula
 
formula

where k is the vertical unit vector. To extract the purely rotational component of a two-dimensional force, we therefore set the Laplacian of the force function equal to the corresponding term in the vorticity equation, solve for the force function, and substitute into (10).

c. Decomposition in three-dimensional flows

In three-dimensional flows, such as considered in section 5, the rotational force can be described in terms of a vector potential AF such that

 
formula

The gradient of any scalar can be added to AF without modifying Frot, and hence it is necessary to prescribe a gauge condition on the divergence of AF. The most convenient choice is

 
formula

in which case the elliptic equation for the vector potential is

 
formula

with the boundary condition

 
formula

Rather than solve (14) directly for AF, it is often more practical to solve for the pressure field through (11b), which still holds in three dimensions, and to infer the rotational force as a residual. However, in section 6, we present an approximate, direct method of solution for AF that holds in the hydrostatic limit.

3. Barotropic Rossby waves

We first consider barotropic Rossby waves on a β plane (e.g., Vallis 2006). The momentum equation, linearized about a state of rest, is

 
formula

Here, u is the horizontal velocity vector, is the two-dimensional gradient operator, f = f0 + βy is the Coriolis parameter, p is the pressure, and ρ0 is the density. Taking the curl of the momentum equation yields the linearized vorticity equation,

 
formula

The two-dimensional streamfunction ψ is defined to satisfy incompressibility. Note that the only force that can change the vorticity and hence the only force on the right-hand side of the rotational momentum equation is the Coriolis force.

a. Phase propagation

We seek solutions that propagate zonally, with a prescribed meridional structure,

 
formula

where ψ0 is the constant wave amplitude; k and l are wavenumbers in the zonal and meridional directions, respectively; and ω is the frequency. This solution holds provided that the classical dispersion relation for barotropic Rossby waves is satisfied,

 
formula

Note that, if we assume ω > 0, then k < 0, which is consistent with westward phase propagation.

We now provide a physical interpretation of the Rossby phase propagation in terms of the rotational force balance. Following the method described in section 2b,

 
formula

The solution for the Coriolis force function is

 
formula

where we have used the dispersion relation (19). Note that only the variation of the Coriolis parameter with latitude contributes to the rotational Coriolis force (ω = 0 if β = 0). In contrast, the remaining component of the rotational Coriolis force,

 
formula

is purely divergent.

Thus, there is a −π/2 phase shift between the Coriolis force function φ and the streamfunction ψ as sketched in Fig. 2, resulting in westward phase propagation. The role of the Coriolis force in this westward phase propagation is less clear if one considers the full force, directed at a right angle to the velocity vector at each point.

Fig. 2.

Schematic diagram illustrating Rossby phase propagation due to rotational Coriolis force.

Fig. 2.

Schematic diagram illustrating Rossby phase propagation due to rotational Coriolis force.

b. Group propagation

The rotational force balance also gives a physically intuitive mechanism for the propagation of a Rossby wave packet. Consider a wave packet solution,

 
formula

where ψ0(x, t) is a slowly varying wave amplitude (in the sense that |0/dx| ≪ |0|, |0/dt| ≪ |ωψ0|). The elliptic problem for φ is now modified,

 
formula

where the prime is a shorthand for d/dx. The leading-order solution takes the form ( appendix A)

 
formula

where

 
formula
 
formula
 
formula

is the zonal group velocity.

The first component of the force function, φphase, is −π/2 out of phase with the streamfunction and represents the same phase propagation force encountered in section 3a. The second component of the force function, φgroup, is in phase with the streamfunction and can be physically interpreted as a group propagation force. As illustrated in Fig. 3 for a short wave (k2 > l2), the force function φgroup is in antiphase with the streamfunction west of the packet center, acting to decelerate the flow. East of the packet center, the force function φgroup is in phase with the streamfunction, acting to accelerate the flow. The net effect is to propagate the wave group eastward. Similar arguments lead to westward group propagation when the wave is long (k2 < l2).

Fig. 3.

Schematic illustrating the role of the rotational Coriolis force in propagating a packet of short Rossby waves eastward.

Fig. 3.

Schematic illustrating the role of the rotational Coriolis force in propagating a packet of short Rossby waves eastward.

c. Energetics

Finally consider the wave energy budget, which again takes a particularly simple and physically intuitive form when written in terms of the force function. The wave energy, integrated over a single wavelength (within which we assume that the wave amplitude is nearly constant), is

 
formula

The rate of change of wave energy is

 
formula

The phase propagation force is out of phase with the velocity and therefore makes no net contribution to the wave energy equation. In contrast, the group propagation force is in phase with the velocity and modifies the energy according to

 
formula

The conservation equation for wave energy immediately follows,

 
formula

4. Wind-driven circulation

We now diagnose the rotational forces acting on the wind-driven circulation in a homogeneous basin. The momentum equation is

 
formula

Here, τ is the surface wind stress, H is the ocean depth, r is the linear friction coefficient, and A is the lateral friction coefficient. Taking the curl of the momentum equation in (28) gives the barotropic vorticity equation,

 
formula

Advection is written in terms of the Jacobian operator, , and we assume that the wind stress is purely zonal, τ = τ(x)i, where i is the horizontal unit vector and

 
formula

with τ0 = 0.2 N m−2. The basin boundaries are at x = 0, L and y = 0, L, where L = 2000 km. Other parameters are ρ0 = 103 kg m−3, β = 2 × 10−11 m−1 s−1, H = 500 m, r = 10−7 s−1, and A = 200 m2 s−1. Boundary conditions are no-normal flow (ψ = 0) and either free slip (∇2ψ = 0) or no slip (n · ψ = 0).

The vorticity equation in (29) is integrated to a steady state on a finite difference grid of 129 × 129 points, giving a lateral resolution of 15.6 km. The Jacobian is discretized according to Arakawa (1966). Time stepping is with a leapfrog scheme and dissipation terms are backward time differenced for numerical stability. The force functions for the rotational component of the forces in (28) are extracted by solving (11a) using a multigrid solver, with the right-hand side of (11a) set equal to the corresponding term in the vorticity equation in (29).

a. Free-slip gyre

In Fig. 4, we show the steady-state streamfunction ψ and absolute vorticity q = ∇2ψ + βy, where a free-slip lateral boundary condition is applied. The wind stress spins up a subtropical gyre with a Sverdrup interior, an inertial western boundary current, and an inertial recirculation subgyre in the northwestern corner (cf. Veronis 1966; Blandford 1971). The exact coincidence of streamlines and absolute vorticity isolines in the western boundary current and inertial recirculation is prevented by the free-slip condition, anchoring the absolute vorticity isolines to their reference latitude on the boundaries. The force functions for the rotational component of each force on the right-hand side of (28) are plotted in Fig. 5.

Fig. 4.

(left) Streamfunction ψ and (right) absolute vorticity q = ∇2ψ + βy for the steady-state free-slip gyre solution. The contour intervals are 104 m2 s−1 for ψ [equivalent to 5 Sv when multiplied by H (1 Sv ≡ 106 m3 s−1)] and 2.5 × 10−6 s−1 for q.

Fig. 4.

(left) Streamfunction ψ and (right) absolute vorticity q = ∇2ψ + βy for the steady-state free-slip gyre solution. The contour intervals are 104 m2 s−1 for ψ [equivalent to 5 Sv when multiplied by H (1 Sv ≡ 106 m3 s−1)] and 2.5 × 10−6 s−1 for q.

Fig. 5.

Force functions for the rotational forces for the free-slip gyre solution shown in Fig. 4: (top) wind and viscous, (middle) Coriolis and friction, and (bottom) inertial. The contour interval is 10−2 m2 s−2 for the wind, Coriolis, and inertia components; and 0.5 × 10−2 m2 s−2 for the viscous and linear friction components. The rotational forces are directed along the contours, at a strength inversely proportional to the spacing between adjacent contours.

Fig. 5.

Force functions for the rotational forces for the free-slip gyre solution shown in Fig. 4: (top) wind and viscous, (middle) Coriolis and friction, and (bottom) inertial. The contour interval is 10−2 m2 s−2 for the wind, Coriolis, and inertia components; and 0.5 × 10−2 m2 s−2 for the viscous and linear friction components. The rotational forces are directed along the contours, at a strength inversely proportional to the spacing between adjacent contours.

The rotational wind stress accelerates a large anticyclonic circulation throughout the basin. Although the wind stress is zonal, the rotational wind stress contains a meridional component in order to maintain continuity of the accelerated flow. Over the Sverdrup interior of the gyre, the rotational wind stress is almost exactly balanced by an equal and opposite rotational Coriolis force. As stated in section 2, if two terms balance pointwise in the vorticity equation (in this case, the curl of the wind stress and advection of planetary vorticity), then the equivalent two terms balance pointwise in the rotational momentum equation: that is, we have a pointwise “Sverdrup balance” in the rotational momentum equation.

Along the western boundary, the rotational Coriolis force accelerates the boundary current northward. Note, as in section 3, that only the variation of the Coriolis parameter with latitude contributes to the Coriolis force function and hence to acceleration of the western boundary current. This result has been previously obtained by Marshall and Tansley (2001) through an integral vorticity budget and proposed as a mechanism to inhibit separation of the western boundary current through suppression of an adverse pressure gradient along the boundary (also see Munday and Marshall 2005). The overall structure of the rotational Coriolis force is consistent with the Rossby phase propagation force in section 3, acting to propagate the entire gyre westward. Viewed in this manner, the latitudinal variation of the Coriolis parameter is the unambiguous source of western intensification of the gyre (Stommel 1948).1

The rotational inertial force decelerates the western boundary current. This is consistent with the expectation that the Coriolis and inertial forces should oppose each other in an inertial western boundary current. However, note that the rotational wind stress makes a surprisingly large contribution to the acceleration of the western boundary current, in contrast with classical theoretical descriptions (e.g., see Pedlosky 1996).

The inertial recirculation subgyre is normally viewed as being driven by the northward advection of low values of vorticity in the western boundary current (Cessi et al. 1987; Ierley and Young 1988). The rotational inertial force indicates that inertia acts to propagate the subgyre eastward by decelerating the flow to the west and accelerating the flow to the east. One can interpret this eastward propagation as resulting from the reversal of the background vorticity gradient and thus the reversed direction of Rossby phase propagation. This latter result is also consistent with the findings of Marshall and Marshall (1992) for the zonal penetration of the recirculation subgyre when dq/ > 0 (consistent with a reversed background vorticity gradient).

Note that the line integrals of the rotational Coriolis and inertial forces vanish along the boundary,

 
formula

This is consistent with each force acting only to redistribute vorticity within the basin.

With free-slip boundary conditions, the viscous and linear frictional forces are both purely rotational,

 
formula
 
formula

The rotational viscous force lies along isolines of relative vorticity and hence is tightly confined to the western boundary, diffusing momentum into basin, whereas the linear frictional force is directed along streamlines, everywhere decelerating the flow.

b. No-slip gyre

With no-slip boundary conditions, the solution settles into a weak limit cycle. Snapshots of the streamfunction and vorticity are shown in Fig. 6 (cf. Bryan 1963). The force functions for the rotational forces are plotted in Fig. 7.

Fig. 6.

As in Fig. 4, but for no-slip gyre solution.

Fig. 6.

As in Fig. 4, but for no-slip gyre solution.

Fig. 7.

As in Fig. 5, but for the no-slip gyre solution shown in Fig. 6.

Fig. 7.

As in Fig. 5, but for the no-slip gyre solution shown in Fig. 6.

As with the free-slip solution, over the Sverdrup interior there is a pointwise balance between the rotational wind stress and rotational Coriolis force. Likewise, over the western boundary current the rotational wind stress and rotational Coriolis force accelerate the flow northward, partially opposed by the rotational inertial force.

The main difference with the free-slip solution is in the rotational viscous force, which is both strengthened and no longer confined to the western boundary region. The increased magnitude of the rotational viscous force can be attributed to the no-slip boundary condition increasing the strength of the viscous force at the western boundary. The increased spatial extent of the rotational viscous force is more subtle. Although the viscous force can still be written as a curl, its component normal to the boundary no longer vanishes. Thus, the rotational net rotational viscous force includes an additional contribution from basin-wide pressure gradients, established to satisfy the kinematic boundary condition and extending the influence of the rotational viscous force throughout the basin.

5. Planetary geostrophic flow

We now turn to planetary-scale baroclinic motion in an ocean of uniform depth. In this section, we consider a simple baroclinic jet with no complications from boundary conditions. In section 7, we will show how the results discussed in this section break down for buoyancy gradients along a solid boundary, leading to an overturning circulation.

The Rossby number is assumed sufficiently small that the inertial acceleration can be neglected. The momentum equation is thus

 
formula

where b is buoyancy. The horizontal and vertical components of (30) are statements of geostrophic and hydrostatic balance, respectively. The rotational part of (30) can be written as

 
formula

from which it follows that

 
formula

Thus, for planetary-scale flow in exact geostrophic and hydrostatic balance, the rotational Coriolis and buoyancy forces are equal and opposite.

a. Example: Baroclinic jet

To illustrate this cancellation between the rotational Coriolis and buoyancy forces, we consider a zonal baroclinic jet in a channel with boundaries at y = 0, L and z = −H, 0. The buoyancy field is prescribed as

 
formula

It follows, from thermal wind balance, that the zonal velocity is

 
formula

where the constant of integration has been chosen such that the velocity vanishes at the seafloor. In Fig. 8, we plot the density anomaly [including a background stratification, Δρ/ρ0 = −(b/g) − (5b0z/gH); Fig. 8a] and the zonal velocity [assuming f = f0(1 + y/L); Fig. 8b]. Also plotted are the full buoyancy and Coriolis forces.

Fig. 8.

Analytic solution for a zonal jet in geostrophic and hydrostatic balance: (a) density anomaly Δρ/ρ0 = −b/g − 5b0z/gH (units: b0/g); (b) zonal velocity u (units: b0H/f0L); (c) buoyancy force bk; (d) Coriolis force −fk × u; (e) rotational component of the buoyancy force ; and (f) rotational component of the Coriolis force .

Fig. 8.

Analytic solution for a zonal jet in geostrophic and hydrostatic balance: (a) density anomaly Δρ/ρ0 = −b/g − 5b0z/gH (units: b0/g); (b) zonal velocity u (units: b0H/f0L); (c) buoyancy force bk; (d) Coriolis force −fk × u; (e) rotational component of the buoyancy force ; and (f) rotational component of the Coriolis force .

An exact analytical solution exists for the rotational components of the buoyancy and Coriolis forces. These are equal and opposite, as expressed by the vector potential,

 
formula

The x component of A defines a force function in the yz plane, by analogy with section 2b. This component of the rotational Coriolis and buoyancy forces is plotted in Figs. 8e,f.

The exact cancellation of the rotational Coriolis and buoyancy forces is equivalent to thermal wind balance,

 
formula

This represents the horizontal component of the vorticity equation for planetary geostrophic flow and describes a balance between tilting of planetary vortex tubes and baroclinic production of vorticity (e.g., Pedlosky 1987).

b. Residual of rotational Coriolis and buoyancy forces

Finally, we write down the elliptic equation for the vector potential, Aresidual = ACoriolis + Abuoyancy, describing the residual of the rotational Coriolis and buoyancy forces,

 
formula

Thus, the residual of the rotational Coriolis and buoyancy forces vanishes whenever the flow is in exact thermal wind balance and exact linear vorticity balance. This is the case in the ocean interior whenever planetary geostrophic dynamics are valid. Conversely, a residual rotational force relies on departures from thermal wind balance and/or linear vorticity balance.

6. Depth-integrated and overturning forces

In the hydrostatic limit, the vector potential AF for the rotational forces can be further decomposed into a depth-integrated component and an overturning component. This is useful for three reasons: (i) it greatly simplifies the method of solution for AF; (ii) it makes explicit connections between the hydrostatic primitive equations, in which the vertical acceleration is neglected, and the three-dimensional rotational forces; and (iii) it enables a clean separation between the rotational forces that drive the depth-integrated (mostly wind-driven) circulation and the overturning (mostly buoyancy-driven) circulation.

a. Decomposition of the vector potential

First, we decompose the vector potential into three components:

 
formula

where Ah represents the horizontal component,

 
formula

is the barotropic part of the vertical component of AF, and is the baroclinic part of the vertical component of AF.

Substituting for AF in the gauge condition (13) gives the scaling

 
formula

where D and L are characteristic length scales for the flow in the vertical and horizontal. The horizontal component of the rotational force is2

 
formula

Thus, in the hydrostatic limit, in which the aspect ratio D/L ≪ 1, we can neglect the baroclinic part of the vertical component .

The elliptic problem for the horizontal component can also be approximated as

 
formula

Integrating (14) from the seafloor, z = −H(x, y), to the sea surface, z = 0, gives the elliptic problem for the barotropic vertical component,

 
formula

where is the horizontal Laplacian operator. Thus, the three-dimensional elliptic problem for AF has been reduced to a two-dimensional Poisson equation in (39) and a pair of second-order ordinary differential equations in (38).

b. Boundary conditions

We require that n · × AF = 0 on all solid boundaries or

 
formula

for all contour loops along the bounding surface. In a nonmultiply connected domain, this is satisfied without loss of generality by setting

 
formula

At the surface boundary, assumed to be a rigid lid, we have

 
formula

At the bottom boundary, where

 
formula

we have

 
formula
 
formula

In the hydrostatic limit, this simplifies to

 
formula

For the barotropic vertical component, we set on all lateral boundaries.3

c. Definition of the overturning force functions

Finally, we write

 
formula

The horizontal component of the rotational force is thus

 
formula

that is, the rotational force decomposes into a depth-integrated force that drives the depth-integrated circulation and an overturning force that drives the overturning circulation. Because AOT vanishes at both the upper and lower boundaries, the components of AOT represent force functions for the overturning force in the two vertical planes.

7. Meridional overturning circulation

The cancellation of the rotational components of the Coriolis and buoyancy accelerations discussed in section 5 breaks down at the lateral boundaries of the ocean due to the kinematic boundary condition of no-normal flow. This precludes thermal wind balance normal to the boundary, leading to a residual overturning buoyancy force by (35) and an MOC.

To illustrate this process, we now diagnose the rotational forces driving the MOC in an idealized model calculation. We solve the hydrostatic primitive equations using the Massachusetts Institute of Technology general circulation model (MITgcm; Marshall et al. 1997), in a basin extending from 63°S to 63°N in latitude, 50° in longitude, and 4 km in depth. The surface temperature and salinity are restored to prescribed functions of latitude on time scales of 6 and 2 months, respectively. We apply free-slip conditions at the lateral boundaries and no-slip conditions at the bottom boundary. Because there is no circumpolar channel, convective sinking at high latitudes in the northern basin is balanced by upwelling over the remainder of the basin, supported by diapycnal mixing of strength 3 × 10−5 m2 s−1 (Munk 1966). Eddies are parameterized following Gent and McWilliams (1990) with an eddy transfer coefficient of 103 m2 s−1. Horizontal and vertical viscosities are 1 × 104 m2 s−1 and 1 × 10−3 m2 s−1, respectively. The equations are discretized on a C–D grid (Adcroft et al. 1999) with a lateral resolution of 1°; the vertical is resolved by 19 levels of variable thickness ranging from 40 m at the surface to 360 m at depth.

In Fig. 9, we show the streamfunction for the MOC, the buoyancy at model levels 1 (0–40 m) and 15 (2210–2560 m), and the horizontal velocities at the same levels. The MOC peaks at 15 Sv, with a weaker surface-confined cell of just over 1 Sv in the Southern Hemisphere. The buoyancy gradient is equatorward in both hemispheres in the surface layer but southward in both hemispheres (aside from the north-eastern corner of the basin) in the deep layer.

Fig. 9.

Numerical calculation of the MOC in an interhemispheric basin. (a) Streamfunction for the MOC (Sv); (b) buoyancy (10−2 m s−2) and (c) horizontal velocity at model level 1 (0–40 m); and (d) buoyancy (10−4 m s−2) and (e) horizontal velocity at model level 15 (2210–2560 m).

Fig. 9.

Numerical calculation of the MOC in an interhemispheric basin. (a) Streamfunction for the MOC (Sv); (b) buoyancy (10−2 m s−2) and (c) horizontal velocity at model level 1 (0–40 m); and (d) buoyancy (10−4 m s−2) and (e) horizontal velocity at model level 15 (2210–2560 m).

In Fig. 10, we show the meridional overturning force functions for the rotational Coriolis and buoyancy forces and their residuals, plotted along the western boundary at longitude 0°, and longitudes 1° and 3°E. Along the western boundary, the rotational buoyancy force accelerates both the northern sinking and shallow southern sinking overturning cells, unopposed by the rotational Coriolis force due to the no-normal-flow boundary condition. However, along the sections at 1° and 3°E, the rotational buoyancy force is greatly compensated by the rotational Coriolis force, leaving a very small residual. Thus, from the perspective of the momentum balance, the buoyancy driving force is only effective at accelerating the MOC immediately adjacent to the boundary, opposed mainly by friction. This confinement of the residual rotational buoyancy force to the boundary is consistent with the observation that boundary buoyancy gradients are far more effective at driving downwelling than interior buoyancy gradients (Spall and Pickart 2001). Note also the qualitative similarity between the structure of driving residual force function along the western boundary (Fig. 10) and the MOC streamfunction (Fig. 9).

Fig. 10.

Overturning force functions for the meridional overturning at: (a) longitude 0° (the western boundary); (b) longitude 1°E; and (c) longitude 3°E, as indicated on the upper-left panel. At each longitude, the panels show the force functions for the buoyancy force, Coriolis force, and residual of the buoyancy and Coriolis forces. The contour intervals are 5 × 10−5 m2 s−2 for all panels.

Fig. 10.

Overturning force functions for the meridional overturning at: (a) longitude 0° (the western boundary); (b) longitude 1°E; and (c) longitude 3°E, as indicated on the upper-left panel. At each longitude, the panels show the force functions for the buoyancy force, Coriolis force, and residual of the buoyancy and Coriolis forces. The contour intervals are 5 × 10−5 m2 s−2 for all panels.

In Fig. 11, we show the zonal overturning force functions for the rotational Coriolis and buoyancy forces and their residuals, plotted at latitudes of 30°S and 30°N and along the northern boundary at 63°N. There is almost complete compensation between the rotational buoyancy and Coriolis forces at 30°S and 30°N, preventing the steeply sloping isopycnals from slumping within the western boundary layers. However, along the northern boundary, the rotational buoyancy force cannot be compensated by the rotational Coriolis force, which vanishes, and hence an overturning cell is accelerated along the northern boundary. In a rectangular basin, convective sinking occurs in the northeastern corner of the basin (Marotzke and Scott 1999); hence, the residual zonal overturning force function in Fig. 11a and the residual meridional overturning force function in Fig. 10a can be viewed as driving a single overturning cell along the northern and western boundaries of the basin.

Fig. 11.

Overturning force functions for the zonal overturning at (a) latitude 63°N (the northern boundary); (b) latitude 30°N; and (c) latitude 30°S, as indicated on the upper-left panel. At each latitude, the panels show the force functions for the buoyancy force, Coriolis force, and residual of the buoyancy and Coriolis forces. The contour intervals are 2 × 10−4 m2 s−2 for all panels.

Fig. 11.

Overturning force functions for the zonal overturning at (a) latitude 63°N (the northern boundary); (b) latitude 30°N; and (c) latitude 30°S, as indicated on the upper-left panel. At each latitude, the panels show the force functions for the buoyancy force, Coriolis force, and residual of the buoyancy and Coriolis forces. The contour intervals are 2 × 10−4 m2 s−2 for all panels.

In the ocean, one expects noncompensation between the rotational buoyancy and Coriolis forces to be confined to within a deformation radius of the coastline. In practice, in an OGCM that does not resolve the deformation radius (as is the case here), this noncompensation will occur over the first grid point. Thus, an important question that requires further study is whether the net overturning force is sensitive to the model resolution.

These arguments provide an alternative but entirely complementary view of the MOC to that presented by Munk and Wunsch (1998). The latter argues that the MOC is driven by mechanical, as opposed to buoyancy, forcing. Although this is the case as far as the mechanical energy budget is concerned, the present results point to a more classical buoyancy-driven interpretation of the flow in the sense that rotational buoyancy forces drive a density current, albeit confined to the western margin of the basin. There is no contradiction with the results of Munk and Wunsch, because the rotational buoyancy force relies on a background buoyancy gradient that, in turn, relies on mechanical energy input for its maintenance. Related discussions of the importance of the available potential energy budget in this context are given by Tailleux (2009) and Hughes et al. (2009).

8. Discussion

In this paper, we have revisited the momentum balance of the large-scale ocean circulation. When a force is applied to a Boussinesq fluid, fluid parcels are accelerated both locally by the applied force and remotely by the pressure gradients established to maintain nondivergence of the resultant motion and to satisfy the no-normal-flow boundary conditions. The net acceleration is described by a rotational force, uniquely determined by a Helmholtz decomposition of the applied force into its rotational and divergent components, along with the kinematic boundary condition that the rotational force should vanish normal to solid boundaries.

To instill confidence in the approach, we have applied this decomposition to a number of classical problems: barotropic Rossby waves, barotropic wind-driven circulation, a baroclinic front, and the overturning circulation in an interhemispheric basin. A key advantage of the approach is that, if two terms balance pointwise in the vorticity equation, then the equivalent two terms balance pointwise in the rotational momentum equation. This allows for vorticity descriptions of the dynamics to be readily reinterpreted in terms of rotational force balances. This provides an alternative, though entirely complementary, interpretation of processes such as Rossby phase and group propagation, western intensification, and the forces driving the large-scale overturning circulation.

The new approach can provide new insights into underlying dynamics of existing problems. An obvious example is the contrasting influence of the viscous force with free-slip and no-slip boundary conditions. With free-slip boundary conditions, the rotational viscous force is directed along contours of relative vorticity (for two-dimensional flow) and is confined to western boundary layers. In contrast, with no-slip boundary conditions, relative vorticity is nonzero on the boundaries; the rotational force must include an additional pressure gradient contribution to satisfy the kinematic boundary condition and therefore spans the entire basin. This insight is only apparent on analyzing the rotational momentum balance.

We conclude by discussing a number of applications of the new approach.

a. Diagnostics in ocean circulation models

In the hydrostatic limit, we have shown that the rotational forces can be further decomposed into a depth-integrated horizontal component and two overturning components, each of which is described by scalar force functions. Because the velocity field is also nondivergent and satisfies the same no-normal-flow boundary condition, precisely the same decomposition can be applied (as in section 6) to the velocity field. This is useful as a diagnostic tool for two reasons: (i) it provides a rigorous dynamical approach for decomposing the circulation and associated momentum balances into depth-integrated and overturning components, and (ii) it is far easier to visualize scalar fields (streamfunctions and force functions) than vector fields (velocities and forces). Work is in progress to diagnose the circulation and rotational momentum balance in an OGCM using this approach.

b. Vorticity-potential formulations of ocean circulation models

Decomposition of both the forces and velocity fields into depth-integrated and overturning components opens up the possibility of formulating an ocean circulation model in terms of vorticity and associated potentials. The resultant rotational momentum equations are

 
formula

where and ΨOT are streamfunctions for the velocity field, defined in an analogous manner to the force functions in section 6. An interesting question is whether this approach avoids the pressure gradient errors that can be so problematic for σ-coordinate models over steep topographic slopes (Mellor et al. 1994).

c. Accurate representation of balance in unstructured-mesh ocean models

The approach developed in this paper has already led to a more accurate method of representing geostrophic and hydrostatic balance in unstructured-mesh ocean models (Maddison et al. 2011, manuscript submitted to Ocean Modell.). These issues are particularly acute in models where the mesh is not aligned with the vertical and small residuals in the representation of geostrophic and hydrostatic balance can swamp the residual acceleration and subsequent evolution of the flow.

d. Boundary layer separation

Boundary layers separate (at high Reynolds number) whenever the flow just outside the viscous boundary layer experiences an appreciable deceleration (Batchelor 1969). In a series of papers (Marshall and Tansley 2001; Munday and Marshall 2005; Ambaum and Marshall 2005), we have developed diagnostic integral balances to determine the processes that accelerate or decelerate a boundary layer in the ocean and atmosphere, thus encouraging its continued attachment or deceleration. The key outstanding challenge is to extend these arguments for separation of two-dimensional flow, from either a lateral coastline or bottom orography to separation of three-dimensional flow from a sloping sidewall. The tools developed in section 6 are precisely those needed to tackle this outstanding problem.

e. Eddy–mean flow interaction

Finally, the decomposition of the forces can be applied not only to the instantaneous momentum balance but equally to the time-mean momentum balance and to the net force exerted by mesoscale eddies on the large-scale circulation. This is appealing for two reasons: (i) the nonlocality of eddy forcing of the mean flow is made explicit, and (ii) it provides a scalar diagnostic (“eddy force function”) that is far less noisy than the equivalent terms in the potential vorticity equation (divergence of the eddy potential vorticity flux) and hence is an attractive metric for assessing different eddy closures. A manuscript describing the application of the approach to eddy–mean flow interaction is in preparation (D. P. Marshall and J. Shipton 2011, unpublished manuscript).

Acknowledgments

We are grateful to Chris Hughes for pointing out an inconsistency in section 6 and to David Munday and Laure Zanna for help with setting up the MITgcm calculations. The comments of two anonymous reviewers led to an improved and more focused manuscript. Financial supported was provided by the U.K. Natural Environment Research Council. The computations described in section 7 were performed at the Oxford Supercomputing Centre, University of Oxford.

APPENDIX A

Solution for Rossby Wave Packet

Here, we outline the derivation of the solution for the force function (24) for a Rossby wave packet. We anticipate that the solution takes the form

 
formula

where M (x, t) and N (x, t) are slowly varying functions. Substituting this trial solution into the elliptic equation in (23), we find that the solution holds, provided that

 
formula
 
formula

Neglecting the second derivatives, because |M″| ≪ |k2M| and |N″| ≪ |k2N|, these simplify to

 
formula
 
formula

Taking d(A1)/dxk × (A2) and again neglecting the second derivatives, we find

 
formula

First, substituting for M′ in (A2), it follows that

 
formula

Second, substituting for N in (A3), we find that

 
formula
 
formula

as required.

APPENDIX B

Free-Surface Boundary Condition for AF

At the free surface, the appropriate boundary condition is that the pressure is constant (more generally, a prescribed atmospheric pressure). This leads to the free-surface boundary condition,

 
formula

As a simple example, consider a surface gravity wave in an ocean of infinite depth. The free-surface displacement at some time is

 
formula

and hence

 
formula

where we exploit the fact that |∂η/∂x| ≪ 1. The gravitational force is F = −gk and hence

 
formula

(The remaining components of Agravity need not be considered because the flow is in the xz plane.) The solution is

 
formula

where x0 is a constant of integration and the root that grows exponentially with depth has been rejected. The surface boundary condition is

 
formula

Substituting for η and to determine the remaining constants, we find

 
formula

This solution is sketched in Fig. B1.

Fig. B1.

Solution for the force function for a free-surface gravity wave.

Fig. B1.

Solution for the force function for a free-surface gravity wave.

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Footnotes

1

Note that this argument is distinct from that presented by Pedlosky (1965) in which trapping of short Rossby waves generated at the western boundary by either friction or inertia is used to explain western intensification.

2

Note that the neglected term of O(D2/L2) need not be aligned with k × ∂Ah/∂z. Thus, the notation O(·) should be interpreted as allowing for arbitrary direction in the neglected terms.

3

One issue that we have not discussed is the nature of the boundary condition for the rotational force at the free surface: this is discussed in  appendix B and illustrated for the problem of surface gravity waves.