OCTOBER 2005 F E R R E I R A E T A L . 1891 Estimating Eddy Stresses by Fitting Dynamics to Observations Using a Residual-Mean Ocean Circulation Model and Its Adjoint DAVID FERREIRA,JOHN MARSHALL, AND PATRICK HEIMBACH Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts (Manuscript received 8 June 2004, in final form 11 March 2005) ABSTRACT A global ocean circulation model is formulated in terms of the “residual mean” and used to study eddy–mean flow interaction. Adjoint techniques are used to compute the three-dimensional eddy stress field that minimizes the departure of the coarse-resolution model from climatological observations of temperature. The resulting 3D maps of eddy stress and residual-mean circulation yield a wealth of infor- mation about the role of eddies in large-scale ocean circulation. In eddy-rich regions such as the Southern Ocean, the Kuroshio, and the Gulf Stream, eddy stresses have an amplitude comparable to the wind stress, of order 0.2 N mϪ2, and carry momentum from the surface down to the bottom, where they are balanced by mountain form drag. From the optimized eddy stress, 3D maps of horizontal eddy diffusivity ␬ are inferred. The diffusivities have a well-defined large-scale structure whose prominent features are 1) large values of ␬ (up to 4000 m2 sϪ1) in the western boundary currents and on the equatorial flank of the Antarctic Circumpolar Current and 2) a surface intensification of ␬, suggestive of a dependence on the stratification N 2. It is shown that implementation of an eddy parameterization scheme in which the eddy diffusivity has an N 2 dependence significantly improves the climatology of the ocean model state relative to that obtained using a spatially uniform diffusivity. 1. Introduction inverse technique to estimate the subgrid-scale fluxes that bring the model into consistency with observations. The parameterization of the transfer properties of The model is formulated in terms of the meteorolo- geostrophic eddies remains a primary challenge and de- gist’s “residual mean” (Andrews and McIntyre 1976), mand of large-scale ocean circulation models used for appropriately modified for application to ocean circu- climate research. The eddy scale contains most of the lation as described in section 2. Eddy terms appear pri- kinetic energy of the global ocean circulation and yet is marily as a forcing of the residual-mean momentum not resolved by coarse-grained models and must be equation—a vertical divergence of an eddy stress that treated as a subgrid-scale process. The conventional ap- can be interpreted as a vertical eddy form drag. Rather proach is to phrase the problem in terms of an eddy than attempt to parameterize the eddy stresses, in sec- diffusivity that relates the eddy flux of a (quasi- tion 4 we infer their large-scale pattern by using them as conserved) quantity to its large-scale gradient. How- control parameters in an optimization calculation. The ever, the spatial distribution of the diffusivity and even associated residual momentum balance and meridional its sign are uncertain. Here, a new approach to the overturning circulation is discussed in section 5. In sec- problem is pursued: a novel formulation of the govern- tion 6, we interpret our results from the more conven- ing equations is employed along with a sophisticated tional perspective of “eddy diffusivities,” ␬. We find that the ␬ implied by our eddy stress maps are not constant but show strong horizontal and vertical spatial variations. In particular, they are surface intensified Corresponding author address: Dr. David Ferreira, Department (reaching values of several thousand meters squared of Earth, Atmospheric, and Planetary Sciences, Massachusetts In- stitute of Technology, Room 54-1515, 77 Massachusetts Avenue, per second) and decay to near-zero value at depth, sug- 2 Cambridge, MA 02139. gestive of a dependence on the stratification N . Incor- E-mail: [email protected] poration of an N 2-dependent eddy diffusivity in the © 2005 American Meteorological Society Unauthenticated | Downloaded 09/25/21 03:57 AM UTC JPO2785 1892 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 Gent and McWilliams (1990, hereinafter GM) eddy pa- Defining the residual velocities and the residual eddy rameterization scheme leads to improvements in model flux as performance. We conclude in section 7. ϭ ϩ ͑ ͒ vres v v* and 4 ١b · Residual-mean framework vЈbЈ .2 ϭ ͑ ͒ Fres zˆ, 5 It is a great conceptual as well as computational ad- bz vantage to phrase the eddy–mean flow interaction the mean buoyancy Eq. (1) becomes problem in terms of residual-mean theory. In that framework, tracers are advected by residual-mean ve- Ѩb F ϩ S. ͑6͒ · ١b ϭϪ١ · ϩ v locities obtained by stepping forward a residual-mean Ѩt res res momentum equation in which eddy stresses appear as • forcing terms. These eddy stresses can then become the Complications arise near horizontal boundaries: vres nˆ • focus of a least squares calculation in which they are and Fres nˆ (where nˆ is a unit normal to the boundary) Ј Ј • ϭ adjusted to “fit” the model to observations. Adjustment are not necessarily zero even though v b nˆ 0. Fol- of the eddy stress terms results, of necessity, in an adia- lowing Treguier et al. (1997), the domain is divided into batic rearrangement of the fluid maintaining tracer an adiabatic interior where the eddy flux is “skew” (Fres Ӎ properties. We begin, then, by reviewing key elements 0) and a surface layer of thickness hs where eddy of residual-mean theory. fluxes develop a diabatic component as they become parallel to the boundary and isopycnal slopes steepen a. The residual-mean buoyancy equation under the influence of turbulent mixing and air–sea ex- changes. This surface layer encompasses the mixed The formulation of the residual-mean buoyancy layer but also includes a transition layer connecting the equation and the definition of the eddy-induced veloc- base of the mixed layer to the low-mixing region of the ity used here follow from Andrews and McIntyre ocean interior (see Ferrari and McWilliams 2005, (1976) and Treguier et al. (1997). Recently, more so- manuscript submitted to Ocean Modell.). In this surface phisticated formulations have been proposed in an at- layer, the definition of ⌿* is modified by assuming that tempt to better separate diabatic and adiabatic eddy the return flow is spread within the layer:1 buoyancy flux components or improve the treatment of boundary conditions (see Held and Schneider 1999; ␷ЈbЈ uЈbЈ ⌿ ϭ ͩ Ϫ ͪͯ ␮ Ϫ Ͻ Ͻ ͑ ͒ Marshall and Radko 2003; Ferrari and Plumb 2003; Ca- * , ,0 , hs z 0, 7 b b ϭϪ nuto and Dubovikov 2005; among others). However, z z z hs the formulation of Treguier et al. (1997) captures the where ␮ is a function that changes from 0 at the surface ϭ ␮ essential physics and is preferred in the present discus- to1atz hs. Note that and hs can be functions of x, sion. y, and t. A similar diabatic layer can be defined at the Assuming a separation between a mean (denoted by bottom of the ocean. Therefore, vres and Fres remain an overbar) and an eddy (denoted by a prime) fields, unchanged in the interior, but they satisfy a non- the mean buoyancy budget is given by normal-flux condition at the top and bottom; more pre- cisely, ⌿* ϭ 0 at all boundaries. Ѩb vЈbЈ ϩ S, ͑1͒ Equation (6) underscores that in a turbulent ocean it · ١b ϭϪ١ · ϩ v Ѩt is vres and not v that advects b. At equilibrium, advec- where S are buoyancy sources (surface forcings and ver- tion by the residual circulation balances the diabatic tical diffusivity). Since eddy buoyancy fluxes tend to be terms. Away from the surface, the diabatic sources S skewed (directed along b surfaces) in the ocean inte- and the eddy diapycnal component Fres decrease, and Ӎ ١ rior, a nondivergent eddy-induced velocity, v*, can be so vres tends to be along b surfaces (vres · b 0; see introduced to capture their effects. The eddy-induced Kuo et al. 2005 for an example). Using the quasi- velocity can be written in terms of a (vector) stream- adiabaticity of mesoscale eddies in the ocean interior, function ⌿*: we simplify the buoyancy Eq. (6) by neglecting the dia- batic eddy flux component F although it may be im- ϫ ⌿*, ͑2͒ res v* ϭϪ١ where 1 Accordingly, the residual flux F is also modified. It is ob- ␷ЈbЈ uЈbЈ res ⌿* ϭ ͩ , Ϫ ,0ͪ. ͑3͒ tained directly using its definition as vЈbЈ minus the skew flux: ١ ϭ Ј Ј ϩ ⌿ ϫ bz bz Fres v b * b. Unauthenticated | Downloaded 09/25/21 03:57 AM UTC OCTOBER 2005 F E R R E I R A E T A L . 1893 portant near the surface (see Marshall and Radko 2003; where ␶ e is given by Eq. (10). Equation (12) retains the Kuo et al. 2005): primitive equation form because we have kept some, but not all, of the O(R ) terms. This is not an issue as Ѩb o ϭ ͑ ͒ long as the O(R ) terms are negligible, which is the case ١ ϩ Ѩ vres · b S. 8 o t in coarse-resolution ocean models such as the one used here (see description in section 3a). However, this does b. The residual momentum equation not hold in general, and Eq. (12) must only be used in Along with the residual-mean buoyancy equation, we appropriate circumstances. must formulate the residual-mean momentum equa- A zonal-average form of Eq. (12) was made use of in tion. It is useful to first consider the momentum equa- Wardle and Marshall (2000) where the eddy terms on tion in the planetary geostrophic limit, in which advec- the rhs were approximated by their quasigeostrophic tion of momentum is neglected entirely: form and related to eddy quasigeostrophic potential vorticity (PV) fluxes.