Mechanical power input from buoyancy and wind to the circulation in an ocean model



[1] We make a systematic quantitative comparison of the effects that surface buoyancy forcing and wind stress have on the energy balance of an idealized, rotating, pole-to-pole ocean model with a zonally re-entrant channel in the south, forced by realistic heat (buoyancy) fluxes and wind stresses representative of global climatology. Surface buoyancy fluxes and wind stress forcing are varied independently; both have significant effects on the reservoirs of various forms of energy and the rates of transfer between them. Importantly, we show for the first time that in the ocean, each power input has a positive feedback on the other. Changes in the rate of generation of available potential energy by buoyancy fluxes at the surface lead to similar changes in the rate of conversion of potential energy to kinetic energy by buoyancy forces (sinking) in the interior, and to changes in the rate of generation of kinetic energy by wind stress. Conversely, changes in the rate of generation of kinetic energy by wind stress lead to changes in the rate of generation of available potential energy by buoyancy forcing. We discuss how this feedback is mediated by the circumpolar current, and processes involving buoyancy, mixing and geostrophic balances. Our results support the notion that surface buoyancy forcing, along with wind and tidal forcing, plays an active role in the energy balance of the oceans. The overturning circulation in the oceans is not the result of a single driving force. Rather, it is a manifestation of a complex and subtle balance.

1. Introduction

[2] Ocean-atmosphere interactions lead to two types of mechanical forcings on the surface of the ocean, namely buoyancy and wind stress. Buoyancy and wind stress are known to play fundamental roles in ocean dynamics [e.g., Gill, 1982]. In contrast, when considering the mechanical energy budget of the ocean, the last decade has seen the role of buoyancy questioned as a source of energy in the circulation, leading to a debate about what processes force, or provide energy for, the global circulation. This debate has been framed largely around the zonally-integrated meridional overturning circulation (the MOC), and is based on the view that “horizontal convection” (i.e., convection due to horizontally variable buoyancy fluxes at a single level, such as the surface of the ocean) cannot generate a deep, turbulent circulation. For example, Munk and Wunsch [1998] argue that only wind and tides are able to supply the energy required to maintain the stratification in the abyssal ocean against the observed rate of deep water formation. Further, Wang and Huang [2005] examine the mechanical energy budget of the ocean (which they define to be the sum of kinetic energy and gravitational potential energy), in which the energy sources are wind work and the work of expansion by buoyancy forcing on the surface, and viscous dissipation is the energy sink. The work of expansion by buoyancy forcing is many orders of magnitude smaller than the energy supplied by wind and tides, so they conclude that buoyancy is not a significant source of mechanical or kinetic energy for the oceans. Many current conceptual modelling efforts continue to be based on these views, with some authors concluding that turbulent mixing generated by winds in fact “drives” the circulation [e.g., Kuhlbrodt et al., 2007].

[3] The notion that convection cannot generate an overturning circulation conflicts with many experiments and numerical simulations [see Hughes and Griffiths, 2006, 2008; Scotti and White, 2011, and references therein]. In numerical simulations combining the effects of wind and buoyancy forcings by Hogg [2010] and Morrison et al. [2011], buoyancy has a significant effect on the dynamics and transport of the MOC and the ACC.

[4] The rate of work by forces that lead to expansion/contraction may be small [Wang and Huang, 2005], but the work by buoyancy forces on the circulation is represented by the rate of conversion of gravitational potential energy (GPE) to kinetic energy (KE). In horizontal convection, differential surface buoyancy forcing generates available GPE (APE), which is converted to KE by the action of buoyancy forces that lead to sinking, and thereby provides energy to a circulation [Hughes et al., 2009; Tailleux, 2009]. In mechanical terms, forced density differences on the surface of the ocean lead to buoyancy forces which drive the circulation. Hughes et al. [2009] show that the conversion rates between energy reservoirs (rather than the amount of mechanical energy in its various potential and kinetic forms) are most relevant to the circulation.

[5] Toggweiler and Samuels [1998], Gnanadesikan et al. [2005] and Gregory and Tailleux [2011] calculate a C, the steady-state net volume-integrated rate of conversion from KE to APE of the resolved flow, from a number of coarse resolution global ocean models with realistic forcing conditions and parameterization choices. According to their calculations C > 0, and Toggweiler and Samuels [1998] and Gnanadesikan et al. [2005] suggest that winds drive the overturning circulation while buoyancy variations on the surface of the ocean are not an active or significant driver of the circulation. In addition, Gregory and Tailleux [2011] estimate CV, the horizontal distribution of the vertically integrated contributions to C, in two global ocean model simulations. They show that in regions associated with deep and bottom water formation and on the western boundaries of the world oceans CV is negative (i.e. net conversion from APE to KE), and that other mechanisms exist that make a contribution to C of similar magnitude but opposite sign (i.e., net conversion of KE to APE).

[6] To date there are no systematic comparisons of how surface buoyancy forcing and wind stress interact or the extent to which they each affect the energetics of ocean circulation. Furthermore, the total reservoir of APE, the rate of generation of APE by surface buoyancy forcing and the rates of conversion from KE to GPE and from GPE to KE in the global oceans have yet to be reliably and accurately estimated.

[7] In this letter we present, for the first time, a systematic comparison of the effects of surface buoyancy forcing and wind stress on the power input to the mechanical energy reservoir (defined here to be the sum of kinetic energy and APE) of an idealized, eddy-permitting ocean circulation model, and of the effects of each forcing mechanism on the rates of energy transfer between the different energy reservoirs. We build mainly upon the work by Hughes et al. [2009], Tailleux [2009] and Gregory and Tailleux [2011] and show that: (1) the pathway by which changes in the rate of generation of APE, associated with changes in buoyancy forcing, affect KE and irreversible mixing is through changes in the conversion of APE to KE (changes in the negative value of C) of similar magnitude; (2) there is a feedback between the effects that surface buoyancy forcing and wind stress forcing have on the rates of generation of APE, KE and the rate of conversion of KE to APE. Our results support the notion that surface buoyancy forcing, along with wind and tidal forcing, plays an active role in the energy budget of the oceans, and that the ocean overturning circulation is not the result of a single driving force. In section 2 we describe the model configuration and runs. In section 3 we summarize the energetics framework we use to analyse the model runs. In section 4 controlled variations of each forcing mechanism allow us to examine the contribution of buoyancy and wind forcings to the energy budget of the system, the pathways by which these contributions are made and the effects on the dynamics of circulation. In section 5 we summarise and conclude.

2. Model Configuration

[8] Solutions are found for an idealized ocean on a 40° longitudinal section of a rotating sphere , 70°N to 70°S. Topography is smooth, with a bottom depth of 4000 m, continental shelves in the north and south of 800 m depth, and an 1800 m deep zonally reconnecting channel in the south, as shown in Figure 1a. A linear equation of state as a function of temperature is used as a proxy for buoyancy variations due to both heat and salinity variations. Heat fluxes are prescribed (as opposed to fixing surface temperatures as in Toggweiler and Samuels [1998] and Gnanadesikan et al. [2005]). We define a reference run with heat fluxes (positive out of the ocean, Figure 1b) and meridional wind stress (τ > 0 for westerly winds, Figure 1b), both representative of zonally averaged global climatology. There is a small negative heat flux in the ACC region and larger positive heat fluxes over the polar shelves (the heat flux in the north is three times that in the south). We alter the reference state by varying either the heat fluxes or the wind stress independently. The net heat flux is zero in all cases.

Figure 1.

Model configuration for the reference run (1/3, 1, 1): (a) pseudo-color plot of topography depth as a function of longitude x and latitude y (note distortion of area due to the projection); (b) heat flux out of the ocean surface and wind stress as a function of y.

[9] Simulations are performed using MITgcm [Marshall et al., 1997] with a horizontal resolution of 1/4° and 36 vertical gridpoints, and a 7th order advection scheme, as described in Shakespeare and Hogg [2012]. No eddy parameterization (which destroys APE without converting it to KE) is implemented, in contrast to Toggweiler and Samuels [1998], Gnanadesikan et al. [2005] and Gregory and Tailleux [2011]. Biharmonic viscosity A4 = 1.0 × 1011 m4/s is used to focus dissipation and mixing on to small length scales, with vertical viscosity νz = 1.0 × 10−5 m2/s; diffusivity coefficients are identical to viscosity. The effects of wind-induced mixing near the surface are parameterised by enhancing vertical diffusivity in the upper 200 m. Sensitivity analyses using different parameterizations indicate that our conclusions are not affected by these modelling choices.

[10] For each run we define a forcing configuration (Q and τ) in terms of the reference configuration in Figure 1 using three parameters, (rNS, c, rτ). First we define heat fluxes Q1(x, y) by setting the ratio of southern to northern cooling, rNS = QS/QN, where QS and QN are the peak heat fluxes at 65°S and 65°N, respectively. Net cooling is equal to net heating over the surface (heating from the reference case is unchanged). Then we uniformly scale Q1 by c to obtain Q = cQ1. The third parameter, rτ = τ/τref, uniformly scales the wind stress from the reference configuration.

[11] The reference case shown in Figure 1 is defined by parameters (1/3, 1, 1). We carry out runs with rNS = 1/7, 1/3 and 1 while holding c and rτ constant at 1. In two further runs the heat forcing of the second and third runs defined previously (rNS = 1/3 and 1) was scaled by c = 2 and 1/2, respectively. These five runs are used to explore the effects of varying buoyancy forcing. To examine the sensitivity to wind forcing we scale the wind stress field in the reference configuration with rτ = 1/2, 2 and 3 while keeping heat fluxes fixed. The parameters defining the forcing fields for all runs are summarised in Table 1.

Table 1. Parameters Defining Heat and Wind Forcing and Energy Fluxes and Reservoirs for All Runsa
  • a

    Fluxes in 10−9 W/kg, KE in 10−3 J/kg and APE in J/kg. The ocean volume is 1.886 × 1017 m3. Run 1 is the reference run. In runs 2–5 and 6–8, the heat fluxes and wind stresses are varied, respectively, keeping all other forcings constant.


[12] Each case was spun up until the difference between the horizontally averaged temperature at the surface and at the bottom of the domain became steady. Spin up times depended on the forcing fields used, but typical values were 2000 years.

3. Energetics Framework

[13] We calculate the energy budget of the mean and turbulent (fluctuating) components of a flow solution, as in Hughes et al. [2009]. We integrate over the ocean volume (V) to calculate the steady state mean kinetic energy

display math

and the available GPE

display math

where u(x, y, z, t), v(x, y, z, t), w(x, y, z, t) and ρ(x, y, z, t) are the computed (resolved) zonal, meridional and vertical velocity and density fields, respectively. 〈〉 denotes a time average, ρ0 is the reference density, M is the mass of water in the domain and g is the gravitational acceleration. The quantity z*(x, y, z, t) represents the depth to which a parcel of water would sink adiabatically under the effect gravitational forces alone. The steady state rates of energy generation and transformation of interest are the rate of generation of APE by surface buoyancy forcing,

display math

the rate of generation of KE from wind stress,

display math

and the rate of conversion of KE into APE (C > 0) by the resolved flow

display math

where α is the thermal expansion coefficient and Cp is the specific heat capacity. C(KE, APE) quantifies the net work done by the resolved flow in the presence of a gravitational field. The transfer C(KE, APE) is offset by conversions by parameterized unresolved buoyancy fluxes.

4. Results and Discussion

[14] Figure 2a shows the mean overturning streamfunction (evaluated on density coordinates) for the reference case. Circulation in this model is dominated by sinking in the form of abyssal water formation at southern latitudes, as discussed in Shakespeare and Hogg [2012], despite buoyancy fluxes being threefold greater in the northern part of the domain. The model's circumpolar current acts as a thermal barrier, isolating the cold waters over the shelf south of 60 °S from the warmer waters to the north, as occurs at the sites of deep water formation in the Ross and Weddell Seas [Orsi et al., 1999]. The spatial distribution of z* for the equilibrium state in the reference run (Figure 2b) shows that the largest contribution to G(APE) comes from the region over the southern shelf, south of 67 °S, where both z* and Q are large in magnitude. The continuous supply of G(APE) over this shelf sustains sinking of heavy abyssal waters, as indicated by the region with values of z* < −3 km in Figure 2b. Sinking in the north, where there is a larger surface heat flux but no zonal reconnection, has a similar but much weaker effect. In the model, the depth of the cell carrying deep water formed in the north is shallower than the North Atlantic Deep Water cell.

Figure 2.

(a) Mean meridional overturning circulation (evaluated on density surfaces); and (b) snapshot of z* in the statistical equilibrium state, (at x = 20 °E) for the reference run (1/3, 1, 1) on a y-z cross section.

[15] Enhancing cooling in the south by increasing QS and decreasing QN (increasing rNS while holding c and rτ fixed) causes G(APE) to increase (Figure 3a, black symbols). Adjacent to the cooling region in the south, isopycnals become steeper and baroclinic instability is enhanced. The meridional density gradient also increases, thereby strengthening the ACC via the thermal wind relation [Borowski et al., 2002; Hogg, 2010; Shakespeare and Hogg, 2012]. The response of the ACC to increased southern buoyancy fluxes results in a positive feedback on the ocean energy cycle: wind work is enhanced resulting in greater generation of kinetic energy (Figure 3c). The increase in G(APE) associated with increased buoyancy forcing over the southern shelf leads to larger mean downwelling velocities and enhanced conversion from APE to KE (Figure 3e).

Figure 3.

Dependence of the power terms G(APE) by buoyancy and G(KE) by wind work, and the conversion C(KE,APE) by the flow, (a, c, and e) on the peak heat flux QS in the south, and (b, d, and f) upon the peak wind stress τ. Symbols are keyed to parameters (rNS, c, rτ) defining each run as shown in Figures 3a and 3b. Note the difference in the ordinate scales between rows of panels.

[16] Uniformly scaling the entire heat flux distribution up with c = 2 or down with c = 1/2 (Figure 3, black square to red square and black star to red star, respectively) causes a greater change in G(APE) compared to changes caused by varying rNS only (equation (3)). As a result of the increase in G(APE), negative C(KE, APE) is enhanced as the downwelling velocities in the north and south both increase (equation (5)). The topographic blockage of zonal flow in the north means there is no feedback like the one mediated by the ACC in the south, and as a result there is no noticeable change in G(KE) associated with either this scaling or that obtained by changing the north-south heat flux ratio with rNS.

[17] Strengthening the wind field enhances the rate of wind working G(KE), as expected (Figure 3d). However, increased Ekman pumping steepens isopycnals in the southern latitudes, resulting in higher z* in the regions of cooling. The ACC is also enhanced by a strengthened wind field, further isolating the cold waters over the southern shelf. This results in another energetic feedback on the system by enhancing the generation of APE from surface buoyancy forcing, G(APE) (Figure 3b). In addition, enhanced G(KE) leads to an increase in mechanical stirring which also causes G(APE) to rise, as discussed in Tailleux and Rouleau [2010], contributing to the energetic feedback of the system. Finally, as per equation (5), buoyancy fluxes associated with increased Ekman pumping counteract downwelling in cold regions, reducing the net energy conversion from APE to KE. At the largest wind stress used in these runs, the net energy conversion is from KE to APE (positive C(KE, APE)); this is likely to be a result of the use of convective adjustment in the model.

[18] The amount of energy in the reservoirs (Table 1) provide a picture complementary to and consistent with the rates of generation and transfer of energy discussed earlier. As a result of the feedback between the forcing mechanisms, increasing either buoyancy forcing or wind forcing increases both the APE and KE reservoirs in the equilibrium states. The available potential energy reservoir is two orders of magnitude bigger than the kinetic energy reservoir: APE = 0.246 J/kg and KE = 2.59 × 10−3 J/kg for the reference run.

[19] In all of the runs, the rate of energy transfer from APE to KE is significant and comparable to the rate of wind working. In the reference run, for example, C(KE, APE) = −0.298 × 10−9 W/kg and G(KE) = 0.457 × 10−9 W/kg. Also, in all cases using the reference winds or weaker (rτ = 1 or 1/2), the rate of generation of APE by buoyancy forcing is a bigger source of mechanical energy than the rate of wind working: G(APE) = 0.782 × 10−9 W/kg in the reference run. In most of our runs C is negative, indicating a net conversion from APE to KE. In estimates of C for global ocean models by Toggweiler and Samuels [1998], Gnanadesikan et al. [2005] and Gregory and Tailleux [2011] C is positive. This difference in sign is likely due to processes not represented by the linear equation of state used here, and to differences in grid resolution, model parameterizations and ocean geometry. However, we have focused on the response of C to changes in surface buoyancy forcing and wind stress.

[20] The physical mechanism by which an increase in surface buoyancy forcing enhances the power input from wind stress can be summarized as follows. Surface buoyancy forcing enters the system through G(APE) in regions where cooling occurs and water parcels lose buoyancy. As negatively buoyant water parcels sink, APE is converted to KE. A consequence of geostrophy is that the greater meridional density gradient that results from an increased meridional surface buoyancy forcing gradient leads to faster mean flows at the surface, thereby increasing the correlation between surface velocity and wind stress and the power input to KE by wind work. Conversely, greater wind stress can rearrange the stratification, leading to the cooling of colder waters at the surface and to an increase in power input to APE from buoyancy forcing. The intermediary between the power input by these two types of forcing is thus the ocean stratification, which responds in such a way as to generate positive feedbacks between each energy source in this model, as discussed in previous paragraphs.

5. Conclusions

[21] Simplified model calculations serve to illustrate the way in which both buoyancy and wind forcings actively play a coupled role in establishing the energy balance and dynamical state of the oceans, affecting available (gravitational) potential energy, kinetic energy and the pathways between the energy reservoirs. We analyse the energetics of the circulation that results from numerical simulations designed to qualitatively mimic the key active regions of the global oceans. We find that enhancements to the rate of generation of APE associated with changes in surface buoyancy forcing lead to similar increases in the rate of conversion of APE to KE associated with sinking of cold waters at high latitudes, and to increases in the rate of wind work. Similarly, enhancements to the rate of generation of KE associated with changes in wind stress lead to similar changes in the rate of generation of APE associated with surface buoyancy forcing, and in the rate of conversion from KE to APE. Importantly, we show for the first time that in the ocean, power input associated with each forcing mechanism (buoyancy and wind) has a positive feedback on the other, with buoyancy fluxes affecting the energy input from wind stress, and vice versa.

[22] The results suggest that the overturning circulation in the oceans is actively modulated by the feedback between the power input from buoyancy forcing and wind stress. We propose that processes related to sinking, mixing, geostrophic balances and the Antarctic Circumpolar Current mediate this positive feedback. A theoretical analysis of the energy conversion pathways [Hughes et al., 2009] predicts that the globally integrated rate of irreversible mixing must balance the rate of generation of available potential energy. This, together with the results presented here, implies that the total mixing rate will increase with wind stress, with total surface heat flux, and with the proportion of the heat flux lost from the high latitude southern region.


[23] Numerical computations were conducted using the National Facility of the Australian National Computational Infrastructure (NCI) in Canberra. This work was supported by Australian Research Council grants DP0986244 and DP1094542. G.O.H was also supported by Australian Research Council Future Fellowship FT 100100869. Special thanks to Callum Shakespeare for running the numerical simulations. Discussions with W. R. Young and R. Tailleux and comments from A. Gnanadesikan and an anonymous reviewer helped improve the quality of this letter and are gratefully appreciated.

[24] The Editor thanks Anand Gnanadesikan and David P. Marshall for assisting in the evaluation of this paper.