热带海洋学报 JOURNAL OF TROPICAL 2013 年 第 32 卷 第 2 期: 1−14 海洋水文学 doi:10.3969/j.issn.1009-5470.2013.02.001 http://www.jto.ac.cn

Evolution of oceanic circulation theory: From gyres to eddies*

HUANG Rui-xin Department of , Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA

Abstract: Physical oceanography is now entering the -resolving era. Eddies are commonly referred to the so-called mesoscale or submesoscale eddies; by definition, they have horizontal scales from 1 to 500 km and vertical scales from meters to hundreds of meters. In one word, the is a turbulent environment; thus, eddy motions are one of the fundamental aspects of oceanic circulation. Studies of these eddies, including observations, theory, laboratory experiments, and parameterization in numerical models, will be the most productive research frontiers for the next 10 to 20 years. Although we have made great efforts to collect data about eddies in the ocean; thus far, we know very little about the three-dimensional structure of these eddies and their contributions to the oceanic general circulation and . Therefore, the most important breakthrough may come from observations and physical reasoning about the fundament aspects of eddy structure and their contributions to ocean circulation and climate. Key words: eddy; baroclinic instability; eddy parameterization CLC number: P731 Document code: A Serial number: 1009-5470(2013)02-0001-14

mostly confined in the upper ocean; thus, it is 1 Landscape of oceanic circulation intensified and ten times stronger than that in an 1.1 -driven circulation vs thermohaline idealized homogeneous ocean. circulation There are numerous definitions for the Wind-driven circulation is defined as the . Literally, the of circulation driven by surface wind stress alone. thermohaline circulation means it is directly linked Note that wind-driven circulation can exist in a to the density difference due to surface thermal homogeneous ocean. In such a conceptual model (heat flux) and haline (freshwater flux) forcing. ocean, the wind-driven circulation is rather sluggish. Therefore, thermohaline circulation was interpreted In contrast, the thermohaline circulation exists in a as the circulation driven by surface stratified ocean only. There is a main in forcing. In textbooks and papers published before the world , which is often treated as an the end of last century, most theories of interface between the warm/light water in the upper thermohaline circulation were based on this ocean and the cold/dense water in the deep ocean. classical framework. In fact, most people believe The main thermocline is actually a zone of strong that thermal forcing can drive thermohaline vertical density gradient, i.e., it is also a . circulation. This framework may stem from the The main pycnocline in the oceans compresses the common wisdom that the ocean is subject to strong wind-driven circulation to the upper 500−800 m. As surface heating/cooling, and such surface a result, the wind-driven circulation in the world heating/cooling provides a huge amount of thermal ocean with a realistic stratification is a circulation energy, which should be able to put the ocean in

Received date: 2012-01-11; Revised date: 2012-10-22. Editor: SUN Shu-jie Biography: HUANG Rui-xin (1942—)，male, scientist emeritus, theory and numerical modeling of large-scale oceanic circulation. E-mail: [email protected] * I have greatly benefited from discussions with my colleagues during the course of the meso-scale eddy workshop held in the South China Oceanology in spring 2011. In particular, comments from two reviewers helped to improve the presentation of this review paper. 2 热带海洋学报 Vol. 32, No. 2 / Mar., 2013 motions, very much like how a burner in the kitchen developed for the gyre-scale circulation, which has works. This line of reasoning was reflected by the a typical horizontal scale of 500−5000 km and parameterizations used in numerical models. In fact, temporal scale on the order of one year or longer. a common practice in numerical modeling was Major wind-driven gyres exist in the world oceans, specifying the vertical mixing coefficient and including the subtropical gyres in the North Atlantic, running the models for different climate conditions, North Pacific, South Atlantic, South Pacific, and completely ignoring the external sources of Indian Oceans, the subpolar gyres in the North mechanical energy required for sustaining vertical Atlantic and North Pacific Oceans. The mixing in a stratified ocean. However, such a corresponding subpolar gyres in the Southern framework is now challenged by a new paradigm Hemisphere are confined to the edge of the based on mechanical energy balance, which we will due to the existence of the strong ACC. discuss shortly. For an in-depth review of oceanic The historic development of theories and general circulation, the reader is referred to a numerical models for basin-scale wind-driven [1] recently published textbook and the references circulation and thermohaline circulation is outlined cited there. on the left part of Fig. 2. At the end of the 1970s, 1.2 Entering the regime of eddies our understanding of the wind-driven circulation in Motions in the ocean are composed of at least the oceans was based on simple two-dimensional two major forms: waves and currents. One can also models. A major breakthrough took place since the say that motions in the ocean are the manifestation 1980s. With the potential vorticity homogenization of geostrophic turbulence, which consists of by Rhines et al.[3] and ventilated thermocline theory difference sizes of eddy. As far as 600 years ago, Da by Luyten et al.[4], the framework of wind-driven Vinci described motions in our world in forms of gyres in the world oceans was completed. At the many large and small eddies[2]. end of the 1980s, a major breakthrough took place Since the world oceans have an extremely along the line of thermohaline circulation. The small aspect ratio (vertical depth vs the horizontal possibility of multiple solution of thermohaline dimension), our description here is mostly focused circulation was first discussed in the seminar work on the horizontal dimension. The largest eddy in the by Stommel[5]; however, it took more than 20 years world oceans is the Antarctic Circumpolar Current (ACC), existing in the form of the global artery with for the related research to become one of the main a horizontal dimension of 20,000 km. Except the streams, and the major breakthrough is represented [6] ACC, currents in the world oceans exist in forms of by the landmark work by Bryan . different size of gyres and eddies (Fig. 1).

Fig. 1 Landscape of the oceanic circulation Fig. 2 Development of theories and models for the oceanic The theory of the oceanic circulation was first general circulation HUANG Rui-xin: Evolution of oceanic circulation theory: From gyres to eddies 3

One of the most important progresses along eddy-resolving. However, it was soon realized that this line is the creation of new energetic view of such models do not resolve eddies; instead, they oceanic general circulation, which took place at the should be called eddy-permitting model. For a long end of last century. Over the long period of time, studies along this line were limited to such development, the thermohaline circulation was eddy-permitting models. With the rapid theorized as a circulation driven by surface development of computer technology, eddy- buoyancy fluxes, including heat flux and freshwater resolving models with resolution of 0.1 degree have flux. As it is well known, the atmospheric become a realistic tool for oceanography, as shown circulation is primarily driven by thermal forcing in the right part of Fig. 2. The major impetus of from the Sun. The great success in atmospheric studying eddies in the ocean is due to the dynamics led people to think the oceanic tremendous potential impact of eddies on the counterpart of circulation could be explained in a oceanic general circulation. As will be shown in the rather similar way, i.e., the oceanic circulation can next section, eddy kinetic energy may consist more be driven by surface thermohaline forcing. With a than 90% of the total kinetic energy in the ocean; huge amount of thermal energy flux going through thus, it is inconceivable that any theory about the air-sea interface, it seems rather a oceanic circulation without eddies can really work. straightforward reasoning that such a huge amount of energy should be able to drive the oceanic 2 The new era of eddies general circulation. 2.1 Energy scales associated with the Over the past 15 years, a new paradigm of wind-driven gyre thermohaline circulation emerged, e.g., Munk et The importance of eddies in the oceanic al.[7], and Huang[8-9]. This paradigm is based on the general circulation can be explained in many second law of thermodynamics. The basic idea is different ways. The most straightforward way is to that although there is a huge amount thermal energy show how much of the total energy is contained by flux going through the ocean surface, such energy eddies. For the wind-driven gyre in the upper ocean cannot be efficiently converted into mechanical of a semi-closed basin, this can be shown as follows. energy because the ocean is heated and cooled from Note that, however, this argument may not apply to about the same geopotential level. In order to the where no zonal boundary exists. overcome the friction associated with thermohaline Omitting the small contribution associated with circulation there is a need of external source of the vertical velocity, the density of total kinetic mechanical energy, which is due to wind stress and energy is tidal dissipation. The major point of this new theory 11 is that parameterization of sub-grid scale processes ehuvghhhf=+=+ρρ()22 '()/ 2 22 2 k 22xy must be based on the constraint of mechanical 1 2222 energy balance. This new methodology has been ≈ ρ ghhLf''/, (1) 2 y applied in many recent studies of parameterization of the diapycnal diffusivity used in numerical where the overbar indicates the horizontal average models. over the whole basin, u and v are the horizontal With the framework of the basin-scale velocity, g′is the reduced gravity, f is the circulation established, we are now entering the parameter, ρ and h are the mean density and regime of eddies. Due to severe technical thickness of the upper layer where the wind-drive limitations of observations and computer power, our gyre is mostly confined, h' is the thickness understating of the dynamics associated with eddies perturbation, and Ly=1000 km is the north-south remains rather rudimentary. In 1980s, models with length scale of the wind-driven gyre. In this rather low resolution of 1/3 degree were called estimate, the contribution due to the east-west

4 热带海洋学报 Vol. 32, No. 2 / Mar., 2013 gradient is small, so it is neglected; accordingly, the form of eddy potential energy. Similar to the kinetic energy associated with the meridional estimate of mean-flow kinetic energy, the eddy velocity is omitted. kinetic energy can be estimated as The density of available gravitational potential 1 eghhkf≈ ρ ''/2222. (5) energy (GPE) is defined as k,eddy 2 Hence, the ratio of eddy potential energy and 11⎡⎤222 eghhghp =−≈ρρ'''() . (2) mean-flow potential energy is 22⎣⎦⎢⎥ −2 Eep,eddy p,eddy k Therefore, the ratio of these two types of energy is = =≈10 . (6) Eeλ 2 k,eddy k,eddy EeL2 pp==y ≈1000 2 , (3) The ratio of eddy kinetic energy to the mean-flow Eekkλ kinetic energy is where λ =≈gh'/ f 30km is the radius of EE/()100≈≈ kL2 . (7) deformation. Thus, the kinetic energy of the mean k,eddy k,mean y flow in a wind-driven gyre is associated with a huge Therefore, eddy kinetic energy is 100 times of the amount of available potential energy[10]. mean-flow kinetic energy. On the other hand, we can estimate eddy With the advance of satellite altimetry, we have potential energy and kinetic energy. Assume that all now a wealth of data collected over the global available potential energy can be converted into oceans. Most importantly, satellite observations eddy kinetic energy, then the total energy of eddies provide a continuously monitored global view of the should approximately equal to the available eddy activity on the sea surface. By comparing kinetic energy for the mean geostrophic flow and that potential energy, that is, for eddies inferred from satellite altimetry, a clear EE≈ . （4） (4) edd p,mean view of the relation between these two terms −1 The horizontal scale of meso-scale eddies is k . In emerged. As shown in Fig. 3, eddy kinetic energy is general, eddy scale is larger than the deformation about 100 times larger than the mean-flow kinetic radius, k −1 > λ ; thus, most eddy energy is in the energy.

Fig. 3 Ratio of eddy kinetic energy and mean (geostrophic) flow kinetic energy. On average, this ratio is the lower bound because the kinetic energy of the mean flow may be somewhat overestimated due to inaccuracy of the geoid used in estimating the mean flow[11]

However, the ratio of 100︰ 1 is not valid the argument listed above does not apply. In everywhere in the world oceans. As seen from the addition, although the big ratio is verified for most lower part of Fig. 3, the corresponding ratio is on part of the surface ocean in the world, it is not clear the order of 10−20 in the core of the ACC. It is clear whether or not such a ratio is valid for the that, due to the specific of a fast circumpolar subsurface ocean. Thus, much work remains to be current, the mean kinetic energy is rather large and done in the future to verify this energy ratio, maybe HUANG Rui-xin: Evolution of oceanic circulation theory: From gyres to eddies 5 using subsurface measurements obtained from this is sometimes called the Rhines scale. and other instruments. There are now new developments along this Eddies in the ocean can be described in terms line of thought. In particular, the so-called of geostrophic turbulence; thus, common wisdom stationary striations have now generated great may suggest that eddies have no large-scale interest in the oceanographic community. As shown organized structure. However, recent studies in Fig. 4, eddies with similar sign of zonal indicate that eddies in the ocean can have some kind geostrophic velocity are organized into some kind of large-scale structure. In fact, two-dimensional of large- scale structure, called stationary striations. turbulence on a rotating sphere tends to organize They can be seen in forms of zonal bands with wave itself into a zonal jet, as shown in both theoretical lengths on the order of 300−500 km, and with sea- studies and idealized numerical simulations. level amplitude on the order of 1 cm. These features Rhines[12] postulated a theory of the beta-plane can penetrate at least the upper 700 m. There are turbulence. He argued that turbulence on a also time-variant striations, which can be detected beta-plane should organize itself into a zonal jet from both altimetric sea-level anomaly and numerical simulations with high resolution. More with the meridional scale of LU= / β , where Rh updated descriptions can be found through web sites U is the root-mean-square velocity of the jet-like via searching with the key word “striation”. The feature, and β is the meridional gradient of the nature of striation and its contribution to the oceanic Coriolis parameter. At middle latitude, β ≈ general circulation remains unclear, and it is now a −11 −1 −1 2×10 m·s . Assuming U ≈1 m·s , LRh≈200 km, research frontier.

Fig. 4 Stationary striations calculated by high-pass-filtered zonal geostrophic velocity[13]

2.2 Size of eddies in the ocean latitudes have a typical horizontal scale of 5−200 A remark regarding the terminology used in km, and are called meso-scale eddies, for which the atmospheric dynamics and oceanic dynamics: In traditional quasi-geostrophic dynamics applies. The atmospheric dynamics, perturbations ( and next order dynamics is called the submesoscale, anticyclones) have typical horizontal scale on the which has a typical horizontal scale on the order of order of 1000 km, and are called synoptic-scale 0.1 to 5 km. At high latitudes, the corresponding phenomena, for which the traditional deformation radius is much smaller, due to weak quasi-geostrophic dynamics applies. The next order stratification; thus, the corresponding scales for of horizontal scale is called meso-scale, which has a meso-scale eddies and submesoscale eddies are typical scale on the order of 100 km. smaller. Note that meso-scale eddies in the ocean In dynamical oceanography, eddies at mid are the dynamical equivalence of synoptic-scale

6 热带海洋学报 Vol. 32, No. 2 / Mar., 2013 eddies in the . Unfortunately, there is eddies near the equator. On the other hand, the currently a confusion of terminology in atmospheric radius of deformation of the first mode is about 10 and oceanic dynamics. km at high latitudes, especially near the Antarctica. In addition, there is a major difference in the At such high latitudes, one degree in latitude nature of eddies in the atmosphere and in the ocean. (longitude) is about 110 km (60 km); thus, even a Synoptic-scale eddies in the atmosphere are closely model with 1/10 degree resolution cannot really related to thermodynamic forcing; on the other hand, resolve eddies at all. meso-scale eddies in the ocean are primarily Although many eddies may be formed with dynamically controlled. their initial radius near the radius of the first The most important horizontal scale often used baroclinic mode, through eddy-eddy interaction, in dynamic analysis is the Rossby radius of their size grows. Since the oceanic general deformation, in particular the radius of deformation circulation takes place in a rather thin layer of fluid, of the first baroclinic mode. A simple way to with a typical aspect ratio of 1000︰1, turbulence in estimate this scale is based on the so-called reduced such fluid environment tends to share some features gravity model. As introduced in Eq. (3), the radius common to two-dimensional turbulence. One of the of deformation is defined as major features of two-dimensional turbulence is the λ = g '/hf, (8) upscale transfer of energy. In fact, many eddies

observed in the ocean are with size on the order of where gg'/=Δρ ρ0 is called the reduced gravity, g 100 km. This is sometimes called the upscale is the gravity acceleration, Δρ is the density cascade of eddies, i.e., part of the eddy energy is difference between the upper layer and the lower transferred from small scales to large scales. layer, ρ0 is constant reference density, and h is The smallest eddies in the ocean are on the the mean thickness of the upper layer. Kolmogorov scale, which is defined as the scale at For a continuously stratified ocean, the which energy dissipation is balanced by molecular corresponding value can be calculated by solving dissipation. From dimensional analysis, the the following eigenvalue problem dimension of dissipation and viscosity is d ⎛⎞f 2 dF [ε]=m2·s−3, [ν]= m2·s−1; thus, the dissipation scale l ⎜⎟n +=λ F 0 , (9) k ddzz⎜⎟2 nn ⎝⎠N 3 1/4 is lk = (νε/ ) . For the world ocean, the total subject to boundary conditions: d/d0,Fzn = amount of external mechanical energy is E=0.1−0.8 zH=−0, , where Ng=−∂ρ / ∂ zis the buoyancy θ TW, and the total volume is V=1.3×1018 m3; so that frequency, ρθ is potential density, Fn is the eigen the mean energy dissipation rate is about function, and z=0 and z=−H are the upper and lower ε=1×10−6 W·m−3. Since the molecular diffusivity is boundaries of the ocean. The corresponding radius ν=1×10−6 m2·s−1, the corresponding Kolmogorov −1/2 of deformation is defined as Rd,ii= λ . scale is about 1cm. In zones of strong dissipation, Due to changes in the background stratification such as the mixed layer, it is on the order of 1 mm. and the Coriolis parameter, the size of radius of How we should go along the line of sub-grid deformation varies greatly over the world oceans. scale parameterization? Right now, most basin-scale The global distribution of the radius of deformation models are running with horizontal resolution on of the first mode was discussed by Chelton et al.[14]. the order of 10 km and vertical resolution of 10 m. At mid latitudes, the typical values for the first, In order to avoid parameterization of sub-grid scale second, and third modes are: 38 km, 18 km, and 11 turbulence, we may have to use grid as fine as km. However, near the equator, the radius of defined by the Kolmogorov scale, i.e., one deformation for the first mode is about 240 km; thus, centimeter or less. Thus, we must increase the 6 it is quite easy for a numerical model to resolve horizontal resolution 10 times and vertical HUANG Rui-xin: Evolution of oceanic circulation theory: From gyres to eddies 7 resolution 103 times. Higher horizontal resolution nonlinear eddies are characterized by having corresponding to smaller time step; thus, the total enclosed streamlines; thus, parcels can be trapped increase in computer power is about 1021. inside eddies and move with eddies. According to the famous Moore’s Law in computer Another important feature of strongly nonlinear industrial, computer power doubles every 18 eddies is that they move faster than linear waves. months. As reported in the Time magazine published From this point of view, there may not be a clear-cut in February 2011, there is an indication that boundary between eddies and waves in the ocean. In computer power increases with a double power law, fact, even 300 years ago Newton came forth and something like exp[e0.0287(yr−1946)]; thus, a super made a brave statement that light can be treated computer in 2050 may be able to do such a super either as waves or as particles. Thus, there should job. However, this may be a dream; and before this be no surprise that we can treat most perturbations grand day arrives, we may still have to rely on observed in the ocean in terms of either waves or sub-grid scale parameterization heavily. eddies. Further debate along this line can be found in a recent paper by Early et al.[16]. 3 Eddies or waves The advantage of using wave dynamics is that There has been a commonly debated issue: there are many powerful tools for analyzing waves. What do we see in the ocean, eddies or waves? To In comparison, however, there is no systematic tool answer this question, we can start from the basic for analyzing eddies. Several books provide equation that describes the movement of excellent review for eddy dynamics, including the [17] perturbations in the ocean. The most commonly most classical book by Tuesdell and the recent used equation for describing perturbations is the book by Wu et al.[18]. In fact, in most cases we have potential vorticity equation, which can be written in to deal with eddies individually; unfortunately, this the following form[15] is so far the only way we can deal with eddies.

∂∂⎡⎤22ψψ ∂ ∂⎛⎞f 2 ∂ ψ ∂ ψ ⎢⎥++⎜⎟0 +β =0 (10) 4 Isolated eddies 22⎜⎟ 2 ∂∂∂∂tzzx⎢⎥⎣⎦∂∂xy⎝⎠ N Lacking powerful tools for the description of where Ψ is the streamfunction, β is the meridional eddies, the existing theory of eddies heavily relies gradient of the Coriolis parameter. The suitable on the study of individual eddies, in particular the boundary conditions at the upper and lower so-called isolated eddies. The study of isolated boundaries are eddies may provide insightful knowledge about ∂2ψ ==−=0, atzH , and z 0 . (11) eddies, their structure and their contribution to the ∂∂tz oceanic general circulation. This equation can be solved by separation of The best known example of the so-called variables, and we obtain the classical wave solution isolated eddies is Jupiter’s Great Red Spot, which that has a relation of may have existed for at least 300 years. Blocking in β k the atmosphere is another well-known phenomenon. σ n =− (12) kl22++λ − 2 n In oceanography, warm/cold rings exist where k and l are the horizontal wave numbers and over a vast area; and they survive in the turbulent

λn is the n-th deformation radius. Therefore, what environment for about one year and carry their we call the Rossby waves are essentially the distinctive chemical and biological characters with vorticity waves, i.e., waves in a rotating ocean is them. closely related to the vortex motion. Some of the well-known examples of isolated Of course, when such perturbations grow, they eddies studied include: Modon, dipole, and heton. may eventually develop closed streamlines; and at Modon is an isolated eddy in a simple analytical this stage, they may well be called eddies. Strongly form. Dipole is a pair of eddies distributed

8 热带海洋学报 Vol. 32, No. 2 / Mar., 2013 horizontally next to each other. Heton consists of a include: current meter, floats, and in particular Argo. pair of vortexes with the same strength of potential Tracking with drift buoys vorticity, but with different signs and located at communicating with satellites and acoustical different depths; and they carry a portion of networks have provided some quite interesting buoyancy anomaly. Heton was discussed in details results. Argo program is probably one of the most by Hogg and Stommel[19]and Gryanik et al.[20]. ambitious ocean observation systems. With up to Flierl[21] presented a comprehensive review for more than 3000 Argo floats moving around the isolated eddies in geophysical fluid dynamics. global oceans, this program can provide continuous McWilliams[22] also reviewed the dynamics of coverage of and distributions submesoscale eddies, in particular the so-called within the upper two kilometers of the world Coherent Vortices (SCV). oceans. Tracing eddies through the combination of 5 Tracking meso-scale eddies TOPEX/Poseidon satellite data and XBT data Progress in understanding the roles of eddies in collected by commercial ships going along some the ocean primarily comes from a wealthy of data repeated lines throughout the ocean is a very good collected from the ocean. Among many different way to collect information about the structure of [25] approaches, tracking meso-scale eddies through eddies in the ocean. Roemmich and Gilson altimetry data is probably the most efficient way up combined these two observation techniques and till now. Much progress has been made along this produced a composed map for the vertical structure line, as reported in the literature, e.g., Chelton et of meso-scale eddies in the North Pacific Ocean (Fig. al.[23-24]. Over the past several decades, satellite 5). This figure clearly shows the three-dimensional observations have provided continuously the state structure of the warm-core and cold-core eddies in on the ocean surface in forms of sea-surface the top 800 m of the ocean. One of the remarkable temperature (SST), ocean color, and altimetry data. features is the westward tilting of eddies in the upper The most updated progress is that now sea-surface 400 m, which is consistent with the theoretical salinity can be measured through satellite. prediction that in order to transfer heat poleward, Other means of observing eddies in the ocean eddies must have the right tilt.

Fig. 5 Composite eddy: left panel for a warm-core eddy; right panel for a cold-core eddy. The nearly horizontal contours indicate the mean temperature (contour increment =2℃, with the warmest contour of 26℃); the bowl-shaped contours indicate the meridional geostrophic velocity (cm·s−1); and color shading indicates temperature anomaly (in units of ℃) [25] HUANG Rui-xin: Evolution of oceanic circulation theory: From gyres to eddies 9

Over the past 10 years, a new technique has interior of subtropical gyres would be a desert of emerged. Seismic oceanography may be a very good ecologic activity; however, satellite images reveal a tool for observing the three-dimensional structure of wealthy of biological activity in the interior of eddies in the ocean[26]. Most commonly used subtropical gyres. Such phenomena pose a techniques cannot really provide an instantaneous challenging question: What has gone wrong with three- dimensional structure of eddies. On the other the traditional theory and model simulation, when it hand, seismic oceanography may be able to provide applies to problems related to chemical and us with a clear view of the three-dimensional biological cycles in the ocean? structure of eddies in the oceans. Of course, such a Recent studies on submesoscale phenomena technique is relatively new, and a lot of work has to may have provide an answer of this seemingly be done in order to adapt and modify this technique puzzling question. The essential point is that at to the need of scientific research in physical horizontal scale of submesoscale eddies and fronts, oceanography. the dynamical constraints on vertical velocity imposed by rotation and stratification may break 6 A mismatch of vertical fluxes down. As a result, the vertical velocity can grow According to the traditional quasi-geostrophic much larger than the corresponding bounds set by theory, both rotation and stratification suppress large-scale dynamics. vertical velocity and vertical exchange of tracer. As Accordingly, the traditional quasi-geostrophic a result, for vast parts of the world oceans, vertical dynamics applies at horizontal spatial scale on the velocity is rather small, on the order of 10−7 −10−6 order of 50−100 km; these eddies are the commonly m·s−1. For a long time, there is a puzzle that the new called meso-scale eddies and have vertical scale on the production rate inferred from oxygen utilization and order of 500−1000 m and time scale of a month (Table cycling rate is about one order of magnitude larger 1). than what is predicted by current basin-scale ocean At horizontal scale of 1 km, both the Rossby general circulation models (OGCMs). number and the Richardson number are of order If the simulation of OGCMs were accurate, the one.

Tab. 1 Typical scales for eddies associated with baroclinic instability in the ocean Horizontal scale / km Vertical scale / m Time scale Rossby number QG Baroclinic instability 50 500 month 0.1 Ageostrophic baroclinic instability 1 50 day 1.0

This is the regime for submesoscale eddies, common wisdom, as often cited in published papers which has a vertical scale of 50 m and a time scale discussing ecology and other marine environmental of one day. The dynamics of submesoscale regime is issues, people tended to make simple statement quite different from the quasi-geostrophic dynamics regarding vertical velocity associated with eddies. for the mesoscale eddies. Within the parameter For example, cyclonic (cold core) eddies used to be range, there is ageostrophic baroclinic instability. identified as equivalent to have ; on the Typical examples include mixed-layer instability other hand, anticyclonic (warm core) eddies implies and restratification. In fact, many outstanding . features associated with small eddies, fronts and In reality, vertical velocity is a rather filaments identifiable from satellite observations, complicated issue needing careful examination. For such as ocean color map and SST, can be classified most cases, vertical velocity of an eddy-like feature into the categories of submesoscale phenomena. can change during its life cycle. There has been a misconception of vertical A more accurate approach of calculating velocity associated with eddies. According to the vertical velocity is to apply the ω -equation based

10 热带海洋学报 Vol. 32, No. 2 / Mar., 2013 on the Q-vector. Hoskins[27]gave a brief introduction velocity in the ocean set by the Ekman pumping is to the Q-vector method. Since then, there is much on the order of 0.04 m·s−1 (that means the water progress, especially in the atmospheric literatures. parcel can move up and down for about 15 m·a−1). Recently, the ω -equation has been applied to the Thus, along the edge of eddies, the vertical velocity study of meso-scale eddies in the ocean. According is two-order of magnitude larger than that of the [28] to Giordani et al. , a generalized ω -equation has mean vertical velocity, according to Giordani et the following form, al.[28]. ∂2w f 22+∇Nw ⋅∇ =∇⋅ Q (13) 2 hh() 7 Baroclinic instability ∂z where w is the vertical velocity; the Q-vector is Baroclinic instability theory has been defined as follows, developed over the past half century, and there are QQ=+ Q ++ Q Q + Q (14) many classical papers and text books published th dm tw dag dr where the subscripts th, dm, tw, dag, and dr indicate along this line. The best known book is the [15] the contributions from: Geophysical Fluid Dynamics by Pedlosky . In a) turbulent buoyancy forcing; most cases, baroclinic instability was examined b) turbulent momentum forcing; under the assumptions of steady flow and small c) kinematic deformation; Rossby number. Among many directions of d) thermal wind imbalance deformation; instability study, there are currently two directions e) thermal wind imbalance trend. worth mentioning: the time-dependent baroclinic A typical case is shown in Fig. 6, which instability and the ageostrophic baroclinic includes strong upwelling and downwelling instability. associated with such eddies. For example, both 7.1 Instability of time-dependent mean flow upwelling and downwelling appear along the edge Pedlosky and his colleagues turned their of eddy A1 (an anticyclonic eddy) and eddy C4 (a attention to the instability of time-dependent mean [29] cyclonic eddy). As shown in this figure, the current. For example, Flierl et al. studied a amplitude of vertical velocity can be on the order of modified Philips 2-layer model for a beta-plane channel. When the frequency of the basic flow 6 m·d−1. matches the multiple of the characteristic frequency of the otherwise stable perturbations, the time-dependent flow becomes unstable, and the instability is similar to the case of resonance. 7.2 Mixed-layer baroclinic instability Mixed-layer theories developed over the past were mostly focused on a quasi one-dimensional model; thus, lateral inhomogeneity plays only a minor role through interaction with currents and meso-scale eddies. Some of the recent new studies on mixed layer, however, focused on the

Fig. 6 Daily-mean vertical velocity (colorbar, in units of submesoscale theory of the mixed layer, in which m·d−1) at 200 m, surface dynamic height (contours, in units lateral density stratification gradient plays a critical of dynamic meter) and the daily-mean surface current role in driving the ageostrophic baroclinic (arrows, in units of m·s−1), during the POMME field instability in a weakly stratified fluid. Most [28] experiment . A1 and C4 are anticyclonic and cyclonic importantly, this instability speeds up the eddies, respectively restratification, affects the potential vorticity In comparison, the typical scale of vertical balance and enhances the transports of tracers HUANG Rui-xin: Evolution of oceanic circulation theory: From gyres to eddies 11 within the mixed layer. Stability analysis for the ageostrophic baroclinic instability associated with the mixed layer was performed by Bocaletti et al.[30]. Using mean profiles of stratification and zonal velocity taken from in-situ observations, they solved the eigenvalue problem associated with the instability of the model ocean. The growth rate curve obtained from the calculation is shown in Fig. 7.

Fig. 8 Restratification of the mixed layer and ageostrophic baroclinic instability

8 Major challenges

8.1 Three-dimensional structure of eddies in the ocean Since we have just entered the regime of eddy-resolving ocean, our knowledge about the structure of eddies remains preliminary. Thus, it is one of the top priorities for us to study the structure

Fig. 7 Interior stability with a vertical structure spanned of eddies in the ocean. Most importantly, our study over the whole depth of the model ocean (a); mixed-layer should start from the examination of individual stability with a vertical structure mostly confined within the eddies in the ocean, in particular their upper 50 m of the model ocean (b); and growth rate as a three-dimensional structure. function of wave number (c)[30] 8.2 Observing and better parameterizing

7.3 Restratification of mixed layer eddies Mixed-layer dynamics is one of the key Historical development in meteorology told us components of oceanic circulation and climate. In that, every time model’s resolution increases, the most published studies, the mixed layer is treated in results obtained from the model subjected to the terms of quasi one-dimensional model, in which the same parameterization used previously turn to be lateral inhomogeneity only plays a minor role inaccurate. It is only after we have learned how to through currents and meso-scale eddies. tune these parameterizations under the guidance of As we enter the regime of submesoscale eddies, physics, the results of model simulation can be there is a new impetus to improve the simulation of improved eventually. mixed-layer dynamics. Lateral density stratification With the rapid development of computer gradient plays an important role in driving the technology, numerical models are now running with ageostrophic baroclinic instability in weakly finer and finer scales. Many current numerical stratified fluid, which speeds up the restratification simulations are running with horizontal resolution and enhances the mixed-layer balance in PV on the order of 1−10 km. (For example, see website: (potential vorticity)/tracer, including temperature hycom.org). However, we do not have many and salinity (Fig. 8). Although recent progress along observational data with such fine scales. Thus, this line has shed light on the improvement of although numerical models keep generating a mixed-layer dynamics, further study is needed; in wealthy of outputs with eddies, we are in a dilemma particular, energy from surface waves must be of not knowing what to do with such new numerical included in the model formulation. simulations, i.e., we cannot say what is correct or

12 热带海洋学报 Vol. 32, No. 2 / Mar., 2013 wrong with these outputs. Eventually, numerical progress along this line includes the introduction of results can be accepted only if we have validated eddy transport term, i.e., separating the advection them with real observations. velocity into mean-flow transport and eddy transport. 8.3 Lateral eddy diffusion If we know the mean-flow velocity, the contribution At the early stage, most basin-scale numerical due to mean flow is relatively easy to calculate. models run with quite low horizontal resolution, on On the other hand, diffusion is directly linked the order of a few degrees. At such low resolution, to small-scale processes, including meso-scale most models have difficulty in resolving narrow eddies, turbulence and internal waves. It is obvious boundary currents, such as the Gulf Stream, the that such small-scale processes cannot be resolved Kuroshio, and the ACC. Thus, in order to resolve in basin-scale numerical simulations. these narrow boundary currents with the low To incorporate the dynamical effects of these resolution used in these models, a common practice sub-scale processes into numerical models, different was to lower the horizontal diffusivity used in the kinds of sub-grid parameterization scheme have simulation as much as possible. been postulated for momentum and tracers, As the computer power increases rapidly, more including momentum dissipation in the vertical and and more energetic eddies appear in model quasi-horizontal directions and tracer diffusion in simulations. Apparently, due to the lack of any the vertical and quasi-horizontal directions. physical constraint, the parameterization of lateral These parameterizations are the most critical tracer diffusion remains fairly subjective. As a components for ocean modeling. Due to historical continuation of the common practice, the same reasons, the parameterization of tracer diffusion has criterion of using the lowest possible lateral been treated as a subject disconnected with the diffusivity has been applied in many numerical balance of mechanical energy of the ocean. It is simulations with resolution high enough to resolve only in the last decade people came to realize that eddies. There is, however, indication that eddies the ocean is not a heat engine and external sources seen in such high-resolution simulations may be too of mechanical energy are required for sustaining the energetic. For example, the Kuroshio exhibits a oceanic general circulation, including both the bimodal behavior on irregular interannual to wind-driven circulation and thermohaline decadal time scale. A detailed PV analysis of such circulation. phenomenon indicates that the bimodal behavior is In recent years, parameterization of vertical closely associated with the gain of positive potential (diapycnal) diffusion of tracers in the ocean based on vorticity, i.e., it is a way for the Kuroshio to get rid mechanical energy available from and wind of the extra negative PV obtained from the stress has been advanced rapidly. It is obvious that basin-scale wind stress curl. Recently, high sub-scale lateral eddy diffusion of tracers is also a resolution numerical simulations with 0.1 by 0.1 key component in oceanic numerical simulation. degree resolution indicate that the frequency of such However, the common practice of quasi-horizontal bimodal behavior seen in the model is much higher eddy diffusion of tracers in the ocean remains nearly than the observed. Thus, it may indicate that the unchanged. To my best knowledge, no paper linking lateral diffusion parameterization used in such quasi-horizontal eddy diffusion to GPE of the system numerical simulations be too low, and the system has ever been published. A common approach in tries to get rid of the extra negative PV by forming ocean modeling is to use harmonic or bi-harmonic such large meandering more frequently than diffusion for quasi-horizontal (horizontal, observed (Bo Qiu, personal communication). along- and along-sigma-surface) eddy A close re-examination reveals that transport of diffusion of tracers. In general, quasi-horizontal eddy tracers consists of two components: advection and diffusion is assumed to be spatially uniform and diffusion. Advection of tracers is directly linked to the isotropic, and the corresponding diffusivity has been velocity field averaged at the grid scale. Recent treated as an adjustable constant, which people can HUANG Rui-xin: Evolution of oceanic circulation theory: From gyres to eddies 13 choose within certain range. In fact, thus far there is The amount of mechanical energy in connection no commonly accepted criterion, which can be used with isopycnal diffusion of tracers is a substantial to judge whether a quasi-horizontal diffusion scheme portion of the total mechanical energy budget for the and its corresponding diffusivity have been chosen world oceans. GPE releasing is highly non-uniform appropriately. in space and highly non-isotropic. In fact, most of the According to the common wisdom, water energy releasing is concentrated in the Southern parcels moving along isopycnal surfaces or along Ocean, in particular in connection with the neutral surfaces do not experience buoyancy force. quasi-stationary eddies in the core of the ACC. These This is equivalent to the statement that isopycnal features should be taken into consideration in the stirring is free from buoyancy work, and equivalent subgrid-scale parameterization of lateral diffusion of to zero GPE change. However, a close examination tracers in numerical models of the next generation. reveals that isopycnal stirring is associated with To conclude this short review, it is clear that change of GPE due to the nonlinear equation of we are entering the era of eddy-resolving ocean. state for . In addition, cabbeling associated There are much more to learn before we can claim with subgrid scale turbulent diffusion can release that we understand the oceanic circulation in this GPE from the mean state. new regime.

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