## 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.

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

**F**acting on an incompressible (or Boussinesq) fluid. The momentum equation is

**u**is the fluid velocity,

*p*is pressure,

*ρ*

_{0}is a reference density, and

*t*is time.

**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,

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

**F**

_{rot}.

*p*

**is the component of pressure associated with**

_{F}**F**. The net acceleration due to

**F**is thus

**F**is the sum of the local applied force and the nonlocal pressure gradient forces established to maintain incompressibility.

### b. Definition of force function in two-dimensional flows

*ϕ*

**, is defined by**

_{F}*ϕ*

**= constant along any boundaries to satisfy the boundary condition (5). The absolute value of**

_{F}*ϕ*

**is arbitrary, but in a closed basin it is convenient to set**

_{F}*ϕ*

**= 0 on the boundaries. The rotational force is directed along contours of**

_{F}*ϕ*

**with a strength inversely proportional to the spacing of the contours.**

_{F}**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

**A**such that

_{F}**A**without modifying

_{F}**F**

_{rot}, and hence it is necessary to prescribe a gauge condition on the divergence of

**A**. The most convenient choice is

_{F}**A**, 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

_{F}**A**that holds in the hydrostatic limit.

_{F}## 3. Barotropic Rossby waves

*β*plane (e.g., Vallis 2006). The momentum equation, linearized about a state of rest, is

**u**is the horizontal velocity vector,

**∇**is the two-dimensional gradient operator,

*f*=

*f*

_{0}+

*βy*is the Coriolis parameter,

*p*is the pressure, and

*ρ*

_{0}is the density. Taking the curl of the momentum equation yields the linearized vorticity equation,

*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

*ψ*

_{0}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,

*ω*> 0, then

*k*< 0, which is consistent with westward phase propagation.

*ω*= 0 if

*β*= 0). In contrast, the remaining component of the rotational Coriolis force,

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.

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

### b. Group propagation

*ψ*

_{0}(

*x*,

*t*) is a slowly varying wave amplitude (in the sense that |

*dψ*

_{0}/

*dx*| ≪ |

*kψ*

_{0}|, |

*dψ*

_{0}/

*dt*| ≪ |

*ωψ*

_{0}|). The elliptic problem for

*ϕ*is now modified,

*d*/

*dx*. The leading-order solution takes the form (appendix A)

The first component of the force function, *ϕ*_{phase}, 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, *ϕ*_{group}, 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 (*k*^{2} > *l*^{2}), the force function *ϕ*_{group} is in antiphase with the streamfunction west of the packet center, acting to decelerate the flow. East of the packet center, the force function *ϕ*_{group} 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 (*k*^{2} < *l*^{2}).

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

### c. Energetics

## 4. Wind-driven circulation

*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,

**=**

*τ**τ*

^{(x)}

**i**, where

**i**is the horizontal unit vector and

*τ*

_{0}= 0.2 N m

^{−2}. The basin boundaries are at

*x*= 0,

*L*and

*y*= 0,

*L*, where

*L*= 2000 km. Other parameters are

*ρ*

_{0}= 10

^{3}kg m

^{−3},

*β*= 2 × 10

^{−11}m

^{−1}s

^{−1},

*H*= 500 m,

*r*= 10

^{−7}s

^{−1}, and

*A*= 200 m

^{2}s

^{−1}. Boundary conditions are no-normal flow (

*ψ*= 0) and either free slip (∇

^{2}

*ψ*= 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* = ∇^{2}*ψ* + *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.

(left) Streamfunction *ψ* and (right) absolute vorticity *q* = ∇^{2}*ψ* + *βy* for the steady-state free-slip gyre solution. The contour intervals are 10^{4} m^{2} s^{−1} for *ψ* [equivalent to 5 Sv when multiplied by *H* (1 Sv ≡ 10^{6} m^{3} s^{−1})] and 2.5 × 10^{−6} s^{−1} for *q*.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

(left) Streamfunction *ψ* and (right) absolute vorticity *q* = ∇^{2}*ψ* + *βy* for the steady-state free-slip gyre solution. The contour intervals are 10^{4} m^{2} s^{−1} for *ψ* [equivalent to 5 Sv when multiplied by *H* (1 Sv ≡ 10^{6} m^{3} s^{−1})] and 2.5 × 10^{−6} s^{−1} for *q*.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

(left) Streamfunction *ψ* and (right) absolute vorticity *q* = ∇^{2}*ψ* + *βy* for the steady-state free-slip gyre solution. The contour intervals are 10^{4} m^{2} s^{−1} for *ψ* [equivalent to 5 Sv when multiplied by *H* (1 Sv ≡ 10^{6} m^{3} s^{−1})] and 2.5 × 10^{−6} s^{−1} for *q*.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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} m^{2} s^{−2} for the wind, Coriolis, and inertia components; and 0.5 × 10^{−2} m^{2} 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.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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} m^{2} s^{−2} for the wind, Coriolis, and inertia components; and 0.5 × 10^{−2} m^{2} 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.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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} m^{2} s^{−2} for the wind, Coriolis, and inertia components; and 0.5 × 10^{−2} m^{2} 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.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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*/*dψ* > 0 (consistent with a reversed background vorticity gradient).

### 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.

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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.

*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

### a. Example: Baroclinic jet

*y*= 0,

*L*and

*z*= −

*H*, 0. The buoyancy field is prescribed as

*ρ*/

*ρ*

_{0}= −(

*b*/

*g*) − (5

*b*

_{0}

*z*/

*gH*); Fig. 8a] and the zonal velocity [assuming

*f*=

*f*

_{0}(1 +

*y*/

*L*); Fig. 8b]. Also plotted are the full buoyancy and Coriolis forces.

Analytic solution for a zonal jet in geostrophic and hydrostatic balance: (a) density anomaly Δ*ρ*/*ρ*_{0} = −*b*/*g* − 5*b*_{0}*z*/*gH* (units: *b*_{0}/*g*); (b) zonal velocity *u* (units: *b*_{0}*H*/*f*_{0}*L*); (c) buoyancy force *b***k**; (d) Coriolis force −*f***k** × **u**; (e) rotational component of the buoyancy force

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Analytic solution for a zonal jet in geostrophic and hydrostatic balance: (a) density anomaly Δ*ρ*/*ρ*_{0} = −*b*/*g* − 5*b*_{0}*z*/*gH* (units: *b*_{0}/*g*); (b) zonal velocity *u* (units: *b*_{0}*H*/*f*_{0}*L*); (c) buoyancy force *b***k**; (d) Coriolis force −*f***k** × **u**; (e) rotational component of the buoyancy force

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Analytic solution for a zonal jet in geostrophic and hydrostatic balance: (a) density anomaly Δ*ρ*/*ρ*_{0} = −*b*/*g* − 5*b*_{0}*z*/*gH* (units: *b*_{0}/*g*); (b) zonal velocity *u* (units: *b*_{0}*H*/*f*_{0}*L*); (c) buoyancy force *b***k**; (d) Coriolis force −*f***k** × **u**; (e) rotational component of the buoyancy force

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

*x*component of

**A**defines a force function in the

*y*–

*z*plane, by analogy with section 2b. This component of the rotational Coriolis and buoyancy forces is plotted in Figs. 8e,f.

### b. Residual of rotational Coriolis and buoyancy forces

**A**

_{residual}=

**A**

_{Coriolis}+

**A**

_{buoyancy}, describing the residual of the rotational Coriolis and buoyancy forces,

## 6. Depth-integrated and overturning forces

In the hydrostatic limit, the vector potential **A _{F}** 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

**A**; (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.

_{F}### a. Decomposition of the vector potential

**A**

*represents the horizontal component,*

_{h}**A**, and

_{F}**A**.

_{F}**A**in the gauge condition (13) gives the scaling

_{F}*D*and

*L*are characteristic length scales for the flow in the vertical and horizontal. The horizontal component of the rotational force is

^{2}

*D*/

*L*≪ 1, we can neglect the baroclinic part of the vertical component

*z*= −

*H*(

*x*,

*y*), to the sea surface,

*z*= 0, gives the elliptic problem for the barotropic vertical component,

**A**has been reduced to a two-dimensional Poisson equation in (39) and a pair of second-order ordinary differential equations in (38).

_{F}### b. Boundary conditions

**n**·

**∇**×

**A**= 0 on all solid boundaries or

_{F}^{3}

### c. Definition of the overturning force functions

**A**

_{OT}vanishes at both the upper and lower boundaries, the components of

**A**

_{OT}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} m^{2} s^{−1} (Munk 1966). Eddies are parameterized following Gent and McWilliams (1990) with an eddy transfer coefficient of 10^{3} m^{2} s^{−1}. Horizontal and vertical viscosities are 1 × 10^{4} m^{2} s^{−1} and 1 × 10^{−3} m^{2} 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.

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).

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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).

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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).

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

In Fig. 10, we show the meridional overturning force functions

Overturning force functions ^{−5} m^{2} s^{−2} for all panels.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Overturning force functions ^{−5} m^{2} s^{−2} for all panels.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Overturning force functions ^{−5} m^{2} s^{−2} for all panels.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

In Fig. 11, we show the zonal overturning force functions

Overturning force functions ^{−4} m^{2} s^{−2} for all panels.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Overturning force functions ^{−4} m^{2} s^{−2} for all panels.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Overturning force functions ^{−4} m^{2} s^{−2} for all panels.

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

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

**Ψ**

_{OT}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

*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

*M*″| ≪ |

*k*

^{2}

*M*| and |

*N*″| ≪ |

*k*

^{2}

*N*|, these simplify to

*d*(A1)/

*dx*−

*k*× (A2) and again neglecting the second derivatives, we find

*M*′ in (A2), it follows that

*N*in (A3), we find that

## APPENDIX B

### Free-Surface Boundary Condition for A_{F}

*η*/∂

*x*| ≪ 1. The gravitational force is

**F**= −

*g*

**k**and hence

*A*

_{gravity}need not be considered because the flow is in the

*x*–

*z*plane.) The solution is

*x*

_{0}is a constant of integration and the root that grows exponentially with depth has been rejected. The surface boundary condition is

*η*and

Solution for the force function

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Solution for the force function

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

Solution for the force function

Citation: Journal of Physical Oceanography 41, 5; 10.1175/2011JPO4528.1

## REFERENCES

Adcroft, A. J., C. N. Hill, and J. C. Marshall, 1999: A new treatment of the Coriolis terms in C-grid models at both high and low resolutions.

,*Mon. Wea. Rev.***127**, 1928–1936.Ambaum, M. H. P., and D. P. Marshall, 2005: The effects of stratification on flow separation.

,*J. Atmos. Sci.***62**, 2618–2625.Arakawa, A., 1966: Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part 1.

,*J. Comput. Phys.***1**, 119–143.Batchelor, G. K., 1969:

*An Introduction to Fluid Dynamics*. Cambridge University Press, 615 pp.Blandford, R. R., 1971: Boundary conditions in homogeneous ocean models.

,*Deep-Sea Res.***18**, 739–751.Bryan, K., 1963: A numerical investigation of a nonlinear model of a wind-driven ocean.

,*J. Atmos. Sci.***20**, 594–606.Cessi, P., G. Ierley, and W. Young, 1987: A model of the inertial recirculation driven by potential vorticity anomalies.

,*J. Phys. Oceanogr.***17**, 1640–1652.Gent, P., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models.

,*J. Phys. Oceanogr.***20**, 150–155.Hughes, G. O., A. M. C. Hogg, and R. W. Griffiths, 2009: Available potential energy and irreversible mixing in the meridional overturning circulation.

,*J. Phys. Oceanogr.***39**, 3130–3146.Ierley, G. R., and W. R. Young, 1988: Inertial recirculation in a

*β*-plane corner.,*J. Phys. Oceanogr.***18**, 683–689.Marotzke, J., and J. R. Scott, 1999: Convective mixing and the thermohaline circulation.

,*J. Phys. Oceanogr.***29**, 2962–2970.Marshall, D., and J. Marshall, 1992: Zonal penetration scale of midlatitude oceanic jets.

,*J. Phys. Oceanogr.***22**, 1018–1032.Marshall, D., and C. E. Tansley, 2001: An implicit formula for boundary current separation.

,*J. Phys. Oceanogr.***31**, 1633–1638.Marshall, J. C., C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasihydrostatic, and nonhydrostatic ocean modeling.

,*J. Geophys. Res.***102**, 5733–5752.Mellor, G. L., T. Ezer, and L. Y. Oey, 1994: The pressure gradient conundrum of sigma coordinate ocean models.

,*J. Atmos. Oceanic Technol.***11**, 1126–1134.Munday, D. R., and D. P. Marshall, 2005: On the separation of a barotropic western boundary current from a cape.

,*J. Phys. Oceanogr.***35**, 1726–1743.Munk, W., 1966: Abyssal recipes.

,*Deep-Sea Res.***13**, 707–730.Munk, W., and C. Wunsch, 1998: The moon and mixing: Abyssal recipes II.

,*Deep-Sea Res.***45**, 1977–2009.Pedlosky, J., 1965: A note on the western intensification of the oceanic circulation.

,*J. Mar. Res.***23**, 207–209.Pedlosky, J., 1987:

*Geophysical Fluid Dynamics*. Springer-Verlag, 710 pp.Pedlosky, J., 1996:

*Ocean Circulation Theory*. Springer-Verlag, 453 pp.Spall, M. A., and R. S. Pickart, 2001: Where does dense water sink? A subpolar gyre example.

,*J. Phys. Oceanogr.***31**, 810–826.Stommel, H., 1948: The westward intensification of wind-driven ocean currents.

,*Trans. Amer. Geophys. Union***29**, 202–206.Stommel, H., and A. B. Arons, 1960: On the abyssal circulation of the world ocean—II. An idealized model of the circulation pattern and amplitude in oceanic basins.

,*Deep-Sea Res.***6**, 140–154.Sverdrup, H. U., 1947: Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern pacific.

,*Proc. Natl. Acad. Sci. USA***22**, 318–326.Tailleux, R., 2009: On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat energy controversy.

,*J. Fluid Mech.***638**, 339–382.Vallis, G. K., 2006:

*Atmospheric and Oceanic Fluid Dynamics*. Cambridge University Press, 745 pp.Veronis, G., 1966: Wind-driven ocean circulation. Part II: Numerical solutions of the non-linear problem.

,*Deep-Sea Res.***13**, 31–55.

^{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*(*D*^{2}/*L*^{2}) need not be aligned with **k** × ∂**A*** _{h}*/∂

*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.