interpretation of the propagation of surface altimetric...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Interpretation of the propagation of surface altimetric1

observations in terms of planetary waves and geostrophic2

turbulence3

Ross Tulloch

Center for Atmosphere Ocean Science, Courant Institute of Mathematical4

Sciences, New York University5

John Marshall

Department of Earth, Atmosphere and Planetary Sciences, Massachusetts Institute6

of Technology7

K. Shafer Smith

Center for Atmosphere Ocean Science, Courant Institute of Mathematical8

Sciences, New York University9

R. Tulloch, Center for Atmosphere Ocean Science, Courant Institute, New York University, 251

Mercer St., New York, NY 10012, USA. ([email protected])

J. Marshall, Dept. of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Tech-

nology, 77 Massachusetts Ave., Cambridge, MA 02139, USA. ([email protected])

K. S. Smith, Center for Atmosphere Ocean Science, Courant Institute, New York University, 251

Mercer St., New York, NY 10012, USA. ([email protected])

D R A F T September 11, 2008, 12:19pm D R A F T

X - 2 TULLOCH ET AL.: WAVES, TURBULENCE AND OCEAN ALTIMETRY

Abstract. The interpretation of surface altimetric signals in terms of Rossby10

waves is revisited. Rather than make the long-wave approximation, the horizon-11

tal scale of the waves is adjusted to optimally fit the phase speed predicted by12

linear theory to that observed by altimetry, assuming a first baroclinic mode ver-13

tical structure. It is found that in the tropical band, the observations can be fit14

if the wavelength of the waves is assumed to be large, of order 600 km or so.15

However polewards of±30◦ it is difficult to fit linear theory to the observations:16

the required scale of the waves must be reduced to about 100 km, somewhat larger17

than the local deformation wavelength. It is argued that these results can be in-18

terpreted in terms of Rossby wave, baroclinic instability and turbulence theory.19

At low latitudes there is an overlap between geostrophic turbulence and Rossby20

wave times scales and so an upscale energy transfer from baroclinic instabil-21

ity at the deformation scale produces waves. At high latitudes there is no such22

overlap and waves are not produced by upscale energy transfer. These ideas are23

tested by using surface drifter data to infer turbulent velocities and timescales24

which are compared to those of linear Rossby waves. A transition from a field25

dominated by waves to one dominated by turbulence occurs at about±30◦, broadly26

consistent with the transition that is required to fit linear theory to altimetric ob-27

servations.28


TULLOCH ET AL.: WAVES, TURBULENCE AND OCEAN ALTIMETRY X - 3

1. Introduction

Altimetric observations of sea surface height (SSH) of the ocean show westward propagating29

phase anomalies in all of the major oceans except the Antarctic Circumpolar Current (ACC),30

the Kuroshio and the Gulf Stream, where the propagation is eastward.Chelton and Schlax31

[1996] attempted to understand these observations in terms of linear, first baroclinic Rossby32

waves in a resting ocean and in the long-wave limit. They found that observed phase speeds33

were larger than predicted by theory outside the tropics by as much as a factor of two [see, for34

example, the introduction ofColin de Verdiere and Tailleux, 2005, who review an extensive35

literature on the subject].Chelton et al.[2007] recently observed that SSH variability appears36

to be nondispersive and consistent with the behavior of nonlinear eddies in many regions of37

the world ocean, particularly poleward of25◦, in western basins and in the ACC. Some of38

the discrepancy between the observations and linear theory can be resolved by including mean39

flow and topography [Killworth et al., 1997;Dewar and Morris, 2000;Killworth and Blundell,40

2005;Maharaj et al., 2007], however phase speeds at higher latitudes are apparently still not41

recoverable with linear theory.42

A number of authors adopt the planetary geostrophic approximation (Dewar, 1998, appendix;43

Killworth and Blundell, 1999, 2003;Colin de Verdiere and Tailleux, 2005) and so automatically44

make the long-wave approximation by neglecting relative vorticity. Others have considered45

Rossby basin modes in the quasigeostrophic approximation [Cessi and Primeau, 2001;LaCasce46

and Pedlosky, 2004]. As noted byKillworth and Blundell[2005], all such calculations implicitly47

assume production of waves at the eastern boundary, yet ray tracing calculations through the48

observed hydrography indicate that such waves are generally unable to cross the basin. Instead,49



Killworth and Blundell[2007] investigate the assumption that waves are produced throughout50

the ocean via local forcing by winds, buoyancy exchange or baroclinic instability of the mean51

state; they put this assumption to use by computing the dispersion relation at each lateral position,52

assuming local forcing and horizontal homogenity (i.e. doubly-periodic boundary conditions53

for each1◦ × 1◦ section, the ‘local approximation’).54

Both Killworth and Blundell [2007] and [Smith, 2007] (in a similar analysis) find that the55

oceans are rife with baroclinic instability, occuring at or below the deformation scale, thus56

providing a ready source of energy,“bubbling up” from below, for the waves and turbulence seen57

at the ocean’s surface. Indeed altimeter observations provide evidence of an inverse spectral58

flux of kinetic energy from the deformation scale up to an arrest wavelength of order 500–100059

km, which decreases with latitude but does not scale closely with the deformation scale [Scott60

and Wang, 2005]. Such an inverse cascade1 is possibly the result of nonlinear interactions in61

geostrophic turbulence. The inverse cascade can be arrested or slowed before reaching the basin62

scale by Rossby waves [Rhines, 1975], stratificationN2(z) [particularly if, as in the ocean,N263

is surface intensified — seeFu and Flierl, 1980;Smith and Vallis, 2001] or dissipitave processes64

[Arbic and Flierl, 2004;Thompson and Young, 2006]. It is not yet clear which of these processes,65

if any of them, sets the ultimate arrest scale.66

Rhines[1975] theorized that, because the eddy timescale increases as the spatial scale grows in67

the inverse cascade, a transition will occur at the spatial scale where the eddy timescale matches68

that of Rossby waves with the same wavelength. The transition scale, commonly referred to as the69

Rhines scale, isLR ∼ (2ut/β)1/2, whereut is the square root of the eddy kinetic energy (which,70

in the two-dimensional system considered, is the only energy). It is at this spatial scale, Rhines71

suggested, that the turbulent energy is shunted into either jets or waves, or both, depending on the72



strength and homogeneity of the eddy field. Numerical experiments presented in Rhines’ paper73

demonstrate that, even when the eddies are energetic enough to form jets, Rossby waves may74

also be energized.Vallis and Maltrud [1993] refined the idea of a wave-turbulence crossover75

by noting that while theβ–effect cannot halt the cascade alone, it inhibits energy transfer into a76

dumbbell-shaped region around the origin in wavenumber space, which leads to the generation of77

zonally elongated flow. There is some evidence for zonal jet formation in the ocean [Maximenko78

et al., 2005;Richards et al., 2006], perhaps a signature of the Rhines effect, in addition to the79

observations of waves byChelton and Schlax[1996] andChelton et al.[2007].80

Recent research [Theiss, 2004;Smith, 2004] has suggested that, on the giant gas planets,81

turbulent generation at small scales should result in jet formation in regions equatorward of some82

critical latitude, and a more isotropic eddy field in regions poleward of that critical latitude.Scott83

and Polvani[2007] confirmed that a critical latitude for jet formation does arise in direct numerical84

simulations of forced-dissipitave shallow-water turbulence on the sphere.Theiss[2006] extends85

the idea further by replacingβ with the mean flow-dependent meridional potential vorticity (PV).86

Specifically, he derives a “generalized” Rhines scale, which includes the effect of mean shears,87

and a corresponding critical latitude, polewards of which jets do not form.88

Following on these ideas,Eden[2007] analyzed eddy length scales in the North Atlantic Ocean89

both via satellite altimetry and an eddy resolving primitive equation model. At high latitudes,90

he shows evidence that eddy scales vary with the Rossby deformation radius, consistent with91

Stammer[1997], while at low latitudes eddy scales are consistent with a generalized Rhines92

scale. That is, eddies scale with the smaller of the deformation radius and the Rhines scale, with93

a critical latitude near 30◦N, where the deformation scale is similar to the Rhines scale.94



In this paper we reinterpret sea surface height (SSH) signals in the context of the aforemen-95

tioned theoretical ideas. Specifically, we avoid the issue of jet formation, but posit that below96

a critical latitude baroclinic eddies transform some of their energy into Rossby waves, and that97

these waves dominate the surface height field. At higher latitudes, where Rossby wave frequen-98

cies are too small to be excited by the inverse cascade, the surface height field remains turbulent.99

We investigate this hypothesis as follows. FollowingKillworth and Blundell[2007], we compute100

the local Rossby wave dispersion, but rather than make the long-wave approximation, we adjust101

the horizontal scale of first baroclinic waves to best-fit the observed phase speeds, and thereby102

infer a length scale for the waves. In the tropics the fitted wavelength is close to both the Rhines103

scale and previously observed SSH scales. Outside the tropics, it is either impossible to match104

the observed phase speeds with Rossby wave speeds at any wavelength (probably because linear105

theory is inadequate) or the fitted wavelength lies near the deformation scale. Using surface106

drifter data to estimate the eddy timescale and energy level, we show that at high latitudes the107

turbulent timescale is faster than the Rossby wave timescale, so turbulence dominates, but at low108

latitudes the Rossby wave and turbulent timescales overlap, enabling the excitation of waves by109

turbulence.110

In section 2 we compare first baroclinic Rossby wave phase speeds calculated with theForget111

[2008] ocean atlas (essentially, a mapping of Argo and satellite altimetric data using interpolation112

by the MITgcm) with observed altimetric phase speeds provided by Chris Hughes. We repeat113

the calculations ofChelton and Schlax[1996] andKillworth and Blundell[2005] with mean114

flow and stratification, but ignore topographic slopes.2 We arrive at a conclusion consistent with115

Chelton et al.[2007] – in mid-latitudes, phase speeds predicted by long-wave linear theory are116

typically faster than observed phase speeds. In section 3, where possible, we fit the phase speeds117



predicted by the linear model to observed phase speeds by adjusting the horizontal scale of the118

waves. We obtain a marked meridional variation in the scale of the fitted waves: equatorwards of119

±30◦ the implied scale is large and close to the Rhines scale, polewards of±30◦ the eddies scale120

with the deformation scale. In section 4 we interpret our result via a comparison of turbulent and121

wave timescales. Finally we estimate the critical latitude at which waves give way to turbulence122

by making use of surface eddy velocities from drifter data, provided by Nikolai Maximenko. In123

section 5 we conclude.124

2. Linear Rossby waves

Rossby waves result from the material conservation of potential vorticity (PV) in the presence125

of a mean gradient. As a parcel moves up or down the background mean PV gradient, its own126

PV must compensate, generating a restoring force toward the initial position. The result is a127

slow, large-scale westward propagating undulation of mean PV contours. Mean currents change128

the structure of the waves in two ways: by altering the background PV gradient (sometimes so129

much so thatβ is negligible), and by Doppler shifting the signal. A number of authors [Killworth130

et al., 1997;Dewar and Morris, 2000;Killworth and Blundell, 2005;Maharaj et al., 2007] have131

shown that the straightforward inclusion of the mean thermal-wind currents in the linear Rossby132

wave problem leads to a much closer agreement between the observed phase speeds and theory.133

Here we take an approach closest toKillworth and Blundell[2007] [see alsoSmith, 2007] and134

compute phase speeds in the local quasigeostrophic approximation, using the full background135

shear and stratification in a global hydrographic dataset. Our focus, however, is on attempting136

to fit the linear results to the satellite data and thereby determining the limitations of linear wave137

theory when mean effects are fully included, and characterizing the scale of the waves that are138



consistent with the observed phase speeds. We now briefly outline the approach, relegating139

details to the appendix.140

We assume that the Rossby wave and eddy dynamics of the ocean away from coasts may be141

represented by the inviscid quasigeostrophic equations, linearized about a slowly varying local142

mean velocityU = U(z)x + V (z)y [seePedlosky, 1984],143

qt + U · ∇q + u · ∇Q = 0, −H < z < 0, (1)

whereq = ∇2ψ+(f 2/N2ψz)z is the quasigeostrophic potential vorticity (QGPV),f is the local144

Coriolis parameter,H is the local depth of the ocean,N2(z) = −(g/ρ0)dρ/dz and the eddy145

velocity isu = −ψyx+ψxy. Defining the buoyancy anomaly asb = fψz = −gρ/ρ0, the mean146

buoyancy asB = −gρ/ρ0 (soBz = N2), and assuming a rigid lid and flat bottom, the upper147

boundary condition is148

bt + U · ∇b+ u · ∇B = 0, z = 0. (2)

Further assuming no buoyancy anomalies at the bottom, the lower boundary condition is149

b(x, y, z = −H, t) = 0. The mean QGPV gradient∇Q, including horizontal shear3, is given by150

∇Q =

[Vxx − Uyx +

(f 2

N2Vz

)z

]x +

[β + Vxy − Uyy −

(f 2

N2Uz

)z

]y (3)

and the mean buoyancy gradients are, via thermal wind balance,∇B = fVzx− fUzy.151

Assuming a wave solutionψ(x, y, z, t) = <{ψ(z) exp [i(kx+ `y − ωt)]}, and likewise forq152

andb, one obtains the linear eigenvalue problem,153

(K ·U− ωm) bm = (`Bx − kBy) ψm, z = 0, (4a)

(K ·U− ωm) qm = (`Qx − kQy) ψm, −H < z < 0 (4b)

whereK = (k, `) andψm is themth eigenvector, sometimes called a ‘vertical shear mode’. (The154

hat notation implies dependence on the wavenumberK, which is suppressed for clarity). The155



eigenvaluesωm are the frequencies of the wave solutions; instability ensues when the eigenvalue156

is complex. The problem is discretized in the vertical using a layered formulation; in the157

discretized case, there are as many shear modes as there are layers. The expressions for the158

discrete surface buoyancyb and q in terms ofψ, and other details of the discretization can be159

found in the appendix, and inSmith[2007].160

For a resting ocean (U = 0, implying Bx = By = Qx = 0 andQy = β), equation (4)161

becomes constant coefficient inz and so simplifies to a problem in which the vertical structure162

is independent of wavenumber. Lettingψ(x, y, z, t) = <{Φ(z) exp [i(kx+ `y − ωt)]}, thenΦ163

satisfies the Sturm Liouville problem164

d

dz

(f 2

N2

dΦm

dz

)= −K2

mΦm,dΦm

dz

∣∣∣∣z=0

=dΦm

dz

∣∣∣∣z=−H

= 0, (5)

whereKm is themth deformation wavenumber andK = |K|. The eigenfunctionsΦm are often165

called the ‘neutral modes’. The eigenvalues satisfy the standard Rossby wave dispersion relation166

ωm =−kβ

K2 +K2m

. (6)

The mean velocity and buoyancy fields are computed from the ocean atlas ofForget[2008], as167

described in the Appendix, and these are used to construct the mean buoyancy and PV gradients.168

At each lateral position in the ocean, we then compute the neutral modes and their deformation169

scales from (5), as well asωm andψm from the complete dispersion relation (4). When comparing170

wavespeeds with altimetric observations, we choose the modem such thatψm correlates most171

closely with the first neutral modeΦ1. We denote the zonal phase speed of this mode as172

cR =ωmk.



2.1. Observations of phase propagation from altimetry

In the long-wave (K = 0), resting ocean limit, the dominant first baroclinic mode has a173

westward phase speedβ/K21 . A zonal average of the long-wave phase speed is computed over174

the central pacific (170◦W to 120◦W), and plotted against latitude in Figure 1. Also plotted are [Fig. 1]175

phase propagation observations provided by C. Hughes (2007, personal communication), zonally176

averaged over the same range. Figure 2 shows global maps of phase speed, “wavelikeness” and [Fig. 2]177

wave amplitude from Hughes’ data.4 Speeds at latitudes 20◦S and 20◦N are well captured by the178

classic long Rossby wave solution. However departures are observed at both low latitudes and179

high latitudes. Observed speeds reach a maximum near±5◦. Poleward of20◦ the Rossby wave180

solution diverges from the observations, reaching roughly a factor of two [Chelton and Schlax,181

1996], and eastward propagation in the ACC region is also not captured.182

A global map of the deformation radii used to calculate the theoretical long-wave phase speeds183

in Figure 1 is shown in Figure 3, and was obtained using theForget [2008] atlas. The vertical [Fig. 1][Fig. 3]

184

structure of the first baroclinic normal mode is plotted on the right for selected latitudes at185

150◦W in the Pacific Ocean, color-coded by the color of the x’s and using solid (dashed) lines186

in the southern (northern) hemisphere respectively. The stratification tends to be more surface187

intensified at lower latitudes, whereΦ1(z = 0) tends toward values near 4, and less surface188

intensified at high latitudes, whereΦ1(z = 0) is between 2 and 3.189

2.2. Observations of oceanic currents and QGPV gradients

Figure 4 shows zonal averages of mean geostrophic zonal velocity and the meridional QGPV [Fig. 4]190

gradientQy from theForget [2008] atlas, with a black contour marking zero. Note that the191

QGPV gradient is nondimensionalized by the value of the planetary vorticity gradient at30◦,192

and that colors are saturated in theQy plot. The important point to note is that∇Q is clearly not193



well approximated byβ. The salient features of theQy plot include: (1) the zero crossing at 1194

km depth in the ACC, (2) the near surface features in the low latitudes, (3) the western boundary195

currents near 40◦N (and the zero crossings below them), and (4) the high latitude bottom water196

formation regions. The zero crossing inQy near 1 km depth in the ACC, just below the zonal jet,197

is responsible for significant baroclinically unstable growth and a steering level at depth in the198

eastward phase propagation of the ACC, as reported inSmith and Marshall[2008]. Low latitudes199

appear to contain Charney instabilities, since zero crossings occur near the surfaces. Finally,200

convectively unstable regions in high latitudes tend to have very small buoyancy frequencies,201

resulting in very large QGPV gradients at those locations.202

2.3. Applicability of linear theory

We now consider the effects of includingU and∇Q, as estimated from theForget[2008] atlas,203

in the dispersion relationship (4). SettingK = 0 (i.e. the long-wave approximation), choosing204

the shear moden such thatψm(z) most closely correlates with the first baroclinic modeΦ1(z) on205

the vertical grid5 and zonally averaging over the same longitudes in the central Pacific (170◦W to206

120◦W) results in the phase speeds represented by the solid gray line in Figure 5. The observed [Fig. 5]207

central Pacific phase speeds from Figure 1 are also replotted for comparison. The long-wave [Fig. 1]208

limit predicts speeds which are too fast in low latitudes and typically (but not always) too slow209

in high latitudes. It is pleasing, however, to now observe eastward propagation in the ACC, a210

consequence of downstream advection by the mean current.211

The assumed spatial scale of the waves also affects the predicted phase speeds. The same212

computation described above, but with deformation-scale waves (K = K1x), gives the dashed213

gray line in Figure 5. Assuming the deformation scale as a lower limit for the wavelength of [Fig. 5]214

the observed waves, the solid and dashed lines in Figure 5 bracket the range of values one can [Fig. 5]215



obtain for the phase speed from linear theory. We address this range of possibilities more fully216

in the next section.217

3. Fitting linear model phase speeds to observations

Traditionally, the long-wave approximation has been used when interpreting altimetric signals218

in terms of Rossby wave theory. The influence of horizontal scale on Rossby wave speed219

has largely been neglected, except for calculations assuming uniform wavelengths of 500 km220

and 200 km reported inKillworth and Blundell [2005]. Chelton et al.[2007] argue that the221

propagation of the observed SSH variability is due to eddies rather than Rossby waves, and remark222

that, equatorward of 25◦, eddy speeds are slower than the zonal phase speeds of nondispersive223

baroclinic Rossby waves predicted by the long-wave theory. Here we show, however, that such224

a difference in speed can be accounted for by linear Rossby waves when their wavelengths are225

chosen appropriately.226

Figure 6 shows both the best-fit phase speeds (left) and the wavelengths associated with those [Fig. 6]227

phase speeds (right) for a zonal average from 170◦W to 120◦W in the Pacific (top) and a global228

zonal average (bottom). We require the waves to fit in the ocean, so exclude wavelengths229

2π/K > 4500 km. We have also assumed that the fitted waves have an east-west orientation230

(` = 0). Settingk = ` makes little difference in the fitted wavelength, which is consistent231

with the finding inKillworth and Blundell[2005] of a weak dependence of phase velocity on232

orientation. In the fitted wavelengths plots, the black x’s correspond to individual latitudes, the233

solid gray curve is a[111]/3 smoothing convolution in latitude of the black x’s, and the thin234

black line is the first deformation wavelength. The fitted wavelengths typically lie between235

600 km and 800 km in the tropics out to about 30◦, and are fairly constant throughout these236

latitudes, with little or no dependence on the deformation wavelength. There is a gap in fitted237



wavelength around±40◦ where the linear theory fails to capture the observed phase speeds. At238

high latitudes, the best-fit is obtained assuming scales near the deformation scale. The inability239

to fit the phase speeds at higher latitudes is suggestive that the ‘wave’ signal is not linear in those240

regions. Clearly, though, the inclusion of wavelengths that result from a best-fit of theoretical to241

observed phase speeds results in a greatly improved prediction.242

Figure 7 shows the importance of the planetary vorticity gradientβ relative to the effect of [Fig. 7]243

mean flowU on the mean QGPV gradient∇Q. Using the length scales computed by the best-fit244

algorithm, we plot the phase speeds that result from settingβ = 0 while keeping the observed245

U (thick dashed line), as well as the phase speeds that result from settingU = 0 and∇Q = βy246

(thin dashed line). [The solid gray line and black x’s are the same as those plotted in the upper left247

panel of Figure 6.] The planetary gradientβ is crucial in the tropics, while in the subtropicsU248

becomes increasingly important, particularly from35◦S to20◦S, where the mean shear accounts249

for much of the factor-of-two phase speed error discussed inChelton and Schlax[1996]. At high250

latitudes the Doppler shift caused byU is crucial in capturing the downstream propagation in251

the ACC. Also note that the most unstable baroclinic modes have horizontal scales (not shown)252

of order the deformation scale polewards of about40◦ and do not increase towards the equator.253

Similarly the maximum baroclinic growth rates, and growth rates at our inferred scales, are254

significantly larger at high latitudes than low latitudes. SeeSmith[2007] for details of the linear255

instabilities.256

4. Wavelike and turbulent regimes in the ocean

A plausible interpretation of the results presented in Section 3 is that in low latitudes, baroclinic257

eddies give their energy to linear Rossby waves, whereas at high latitudes Rossby waves are less258

easily generated, and the SSH field remains dominated by eddies. This can be understood in259



terms of a matching, or otherwise, of turbulent and wave timescales, as discussed in the barotropic260

context byRhines[1975] andVallis and Maltrud[1993], and in a (first-mode) baroclinic context261

applied to the gas planets byTheiss[2004], Smith[2004] andTheiss[2006]. The central idea262

of the Rhines effect is that, as eddies grow in the inverse cascade, their timescale decreases, and263

when this timescale matches the frequency of Rossby waves with the same spatial scale, turbulent264

energy may be converted into waves, and the cascade will slow tremendously. When this idea is265

applied to a putative interaction withbaroclinic Rossby waves, there is the added complication266

that frequencies tend toward 0 at large scale (see Figure 8). In this case, only sufficiently weak [Fig. 8]267

eddies have timescales, at any wavelength, that intersect the Rossby wave dispersion curve.268

For illustrative purposes, one can estimate the wavenumber at which the intersection occurs269

by assuming a turbulent dispersion relationship of the formωt = kut, whereut is the turbulent270

velocity scale (the square root of the appropriate eddy kinetic energy). Setting this equal to the271

absolute value of the approximate Rossby wave frequency (ωR ' −kQy/(K2 +K2

1)), we have272

(dividing byk)273

ut ∼Qy

K21 +K2

. (7)

Solving forK gives the relationshipK2 = Qy/ut−K21 , for which there is a real solution only if274

Qy/ut > K21 . At fixedQy andK1, the implication is that waves can be generated (and the cascade275

inhibited) only when the turbulent energy is sufficiently small. On the other hand, assuming a276

constantut, and noting thatQy (through its dependence onβ) andK1 (which is proportional277

to f ) are dependent on latitude, the relationship (7) implies the existence of a critical latitude,278

polewards of which no intersection is possible.279

Let us now see what the data suggests about a relationship like (7). We replace the approx-280

imate Rossby wave dispersion relation with the frequencies from (4), using the fitted Rossby281



wave scales described in the previous section. The idea is illustrated in Figure 8, which shows [Fig. 8]282

zonally averaged Rossby wave frequency curvesωR(k), plotted against zonal wavelength, at283

three latitudes in the tropical Pacific Ocean. Two hypothetical eddy frequency curvesωt = kut284

(dashed lines) are added for comparison, withut = 10 cm s−1 andut = 5 cm s−1. At 10◦S the285

eddy frequency curves intersect the Rossby wave frequencies at relatively small wavelengths,286

indicating that observed tropical SSH length scales are certainly in the wave region. On the other287

hand, at30◦S even the 5 cm s−1 curve fails to intersectωR(k). We thus expect little wavelike288

activity outside the tropics.289

We can improve the frequency comparison test further by using observations of surface drifter290

speeds to obtain estimates ofut. A global map of the root mean square (rms) of the surface291

drifter data292

urms(0) =√|u′drifter(z = 0)|2

(courtesy of Nikolai Maximenko) is shown Figure 9, with its zonal average over 170◦W to [Fig. 9]293

120◦W (the region within the rectangle) plotted on the right. The zonal average is strongly294

peaked at the equator, and more constant at extra-tropical latitudes. However, this may not be295

indicative of the distribution of depth-integrated eddy kinetic energy, since the surface velocity296

gives no information about the vertical structure of eddying motion. Additional assumptions are297

necessary to extract the relevant eddy velocity scale.298

Wunsch[1997] showed that, away from the equator, eddy velocities are primarily first-299

baroclinic, with a smaller projection onto the barotropic mode, while nearer the equator, motions300

tend to have a more complex vertical structure, projecting onto many higher modes, approaching301



equipartition. Expandingurms(z) in the neutral modes (5), we have302

urms(z) =Nz∑m=0

Φm(z)um.

FollowingWunsch[1997], we extract the vertical structure at each location by assuming thatthe303

rms velocity projects entirely onto the first baroclinic mode, which givesu1 = urms(0)/Φ1(0).304

We take the relevant eddy velocity scale to be the root vertical-mean square velocity, thus305

ut =

[1

H

∫ 0

−Hurms(z)

2dz

]1/2

= urms(0)/Φ1(0)

where we have used the orthonormality of the neutral modes.6 The scaling by the first baroclinic306

mode has the effect of reducing the estimated turbulent velocity scale in regions of strongly307

surface intensified stratification, such as near the equator. In these regions, the first baroclinic308

mode itself is quite surface intensified, soΦ1(0) can be considerably larger than one (see the309

modal structure in Figure 3). Physically, if the first neutral mode, onto which all the motion is [Fig. 3]310

assumed to project, is very surface intensified, then eddy velocities are weak at depth, so the311

total turbulent velocity estimate is diminished.312

Figure 10 shows the eddy velocity scaleut and zonal Rossby phase speedcR zonally averaged [Fig. 10]313

over 170◦W to 120◦W and plotted against latitude. These are essentially the left hand and314

equivalent right hand sides of Eq. (7). Our Figure 10 is similar to Fig. 3 ofTheiss[2006] [Fig. 10]315

for Jupiter, except that here our dispersion relation is computed from the full vertical structure316

of the mean flow, rather than just the first baroclinic component (because of the dominance317

of the first baroclinic mode, however, the first baroclinic calculation is rather similar — not318

shown). Note thatut is nearly constant with latitude, varying between and 5 and 10 cm s−1 —319

the strong equatorial values have been reduced, through projection onto the surface-intensified320

first baroclinic mode, as explained above (if one assumed equipartition, the velocity estimate in321



the equatorial region would be reduced even further). In contrast, the (Doppler-shifted) Rossby322

wave speed varies markedly, exceeding 20 cm s−1 in the tropics and falling toward zero at higher323

latitudes (and even becoming prograde in the ACC). The cross-over between the two curves324

occurs at a latitude of roughly±25◦. Note that since we have assumed that the turbulent velocity325

scaleut is entirely in the first baroclinic mode, the crossover latitudes should be considered as326

lower bounds.327

The lower plot in Figure 10 shows the ratio of linear phase speedscR to the eddy velocity scale [Fig. 10]328

ut, with dashed lines denotingcR/ut = 2 and1/2. Theiss[2006] shows that stormy regions on329

Jupiter are highly correlated with regions where this ratio is less than one. Notably,±25◦ is also330

the crossover latitude between linear wavelike behavior and nonlinear eddies found byChelton331

et al. [2007]. Outside this latitude band, first-baroclinic Rossby wave timescales cannot match332

the turbulent timescales implied byut. Note that this would not preclude the formation of the333

mid-latitude zonal jets observed byMaximenko et al.[2005] andRichards et al.[2006]: since334

barotropic Rossby waves are possible, turbulent energy can still accumulate around the dumbbell335

of Vallis and Maltrud[1993].336

Finally we return to a consideration of the spatial scales obtained by fitting linear Rossby337

wave theory to observed phase speeds, as in Figure 6. A global zonal average of the fitted [Fig. 6]338

wavelengths is plotted against latitude in Figure 11. Also plotted are both observed (black o’s) [Fig. 11]339

and simulated (black x’s) eddy wavelengths fromEden[2007], as well as observed wavelengths340

(small circles with line) fromChelton et al.[2007]7. The deformation wavelength (thin black341

line) is also plotted for reference. At low latitudes all of the scales are in close agreement, while342

the fitted wavelength diverges from the observed eddy scales at latitudes poleward of about±30◦.343

This is also the latitude where Eden’s scales transition from a flatter Rhines scaling to a steeper344



deformation scaling. In the Southern Ocean there is a transition from westward propagation to345

eastward propagation upon entering the ACC region. Finally we note that, in contrast toEden346

[2007], Chelton’s data do not exhibit a clear transitional latitude between Rhines scaling and347

deformation scaling. The reasons for this remain unclear.348

5. Conclusions

We have revisited the interpretation of altimetric phase speed signals in terms of linear Rossby349

wave theory. Given observations of the interioru and∇Q fields [courtesy ofForget, 2008],350

and assuming quasi-geostrophic theory, we adjusted the lateral scale of linear waves to best fit351

altimetric observations of westward phase propagation. We find that the implied scales have a352

well-defined meridional structure. In low latitudes the waves have a scale of600 km or so, broadly353

consistent with an appropriately defined Rhines scale. In high latitudes it is more difficult to fit354

linear theory to the observations, but our attempts to do so imply a scale that is much smaller than355

in the tropics, closer to the local Rossby deformation scale. There is a rather abrupt transition356

from low-latitude to high-latitude scaling at±30◦. These results are broadly consistent with357

observed and modeled eddy scales, as reported inEden[2007].358

We put forward an interpretation of the reported results in terms of the interaction between359

turbulence and waves. Over vast regions of the ocean, at scales on or close to the Rossby360

deformation scale, baroclinic instability converts available potential energy to kinetic energy361

of turbulent geostrophic motion. Non-linear interactions result in an upscale energy transfer.362

At low latitudes, where we observe thatut < |cR|, turbulence bubbling up from below readily363

excites Rossby waves. At higher latitudes, whereut > |cR|, turbulence cannot readily excite364

waves because of the weak overlap in timescales between turbulence and waves. Making use of365

surface drifter observations, we estimate that the latitude at which waves give way to turbulence366



coincides with that at whichut ∼ |cR|, and is found to be±30◦ or so, roughly consistent with367

the transition from waves to non-linear eddies recently highlighted byChelton et al.[2007].368

Appendix: Mean State Calculation from the Forget Atlas and Discretization of Linear

Problem

TheForget[2008] ocean atlas contains up to 50 layers (of thicknesses∆j) of potential temper-369

ature and salinity data at each (lat, lon) coordinate. We first compute annually averaged global370

potential temperature and salinity fields, and from these compute a neutral density fieldρ using371

locally referenced pressure. Thermal wind balance is then used to compute the mean velocity372

field U, assuming a level of no motion at the bottom of the ocean [see the appendix ofSmith,373

2007, for details]. We define the top 5 layers, which are each 10 m thick, as a mixed layer of374

depthh ≡ 50 m. The mean buoyancy gradients∇B = −(g/ρ0)∇ρ at the surface are averaged375

over the defined mixed layer, and then related to vertical shears via thermal wind376

Uz(z0) = − 1

fh

∫ 0

−hBydz, Vz(z0) =

1

fh

∫ 0

−hBxdz.

The surface velocities themselves are obtained by averaging the velocities from the ocean atlas377

overh, viz.378

U(z0) =1

h

∫ 0

−hUdz, V (z0) =

1

h

∫ 0

−hV dz. (A1)

The linear problem is discretized, at each lateral location, onto theNz discrete depthszj of379

the data computed from the Forget atlas. The discrete surface buoyancy is given by380

bm(z0) = fψm(z0)− ψm(z1)

∆0

.

and the discrete PV is381

qm(zj) =f 2

∆j

[ψm(zj−1)− ψm(zj)

B(zj−1)−B(zj)− ψm(zj)− ψm(zj+1)

B(zj)−B(zj+1)

]−K2ψm(zj), j = 1..Nz − 1.



The mean QGPV gradientsQx(zj) andQy(zj) are given by (3), using the same vertical dis-382

cretization, and simple horizontal finite differences to computex and y derivatives. At the383

bottom,qm(zNz) = ψm(zNz) = 0. The discrete version of (4) is then solved as a single matrix384

eigenvalue problem, using Matlab.385

Acknowledgments. The authors thank Ryan Abernathy, Gael Forget, Chris Hughes and Niko-386

lai Maximenko for help with, and access to datasets; and Geoff Vallis for a helpful discussion387

on this subject.388

Notes1. The inverse cascade observed byScott and Wang[2005] presented a conundrum since up to 70% of the variability at the ocean surface is

contained in the first baroclinic mode [Wunsch, 1997] and it was thought that first baroclinic energy should cascade towards the Rossby

radius. HoweverScott and Arbic[2007] showed in simulations of two-layer baroclinic turbulence, that while the total energy in the first

baroclinic mode cascades towards the Rossby radius, the kinetic energy moves upscale.389

2. Tailleux and McWilliams[2001] argued that “too fast” Rossby waves may be a consequence of them not ‘feeling’ the bottom of the ocean, an

effect that may emerge when topographic slopes exceed theβ-effect [see the discussion inColin de Verdiere and Tailleux, 2005].Killworth

and Blundell[1999], however, include topographic slopes in their calculation and note that even though topographic wave speeds can vary

between 0.4 and twice the local flat-bottom phase speed, the effects largely cancel over the basin, and so are unlikely to affect phase speeds

after zonal averaging. Note also that Fig. 6 ofMaharaj et al.[2007] indicates that such topographic slopes contribute little to the dispersion

relation at large scales.

3. Horizontal shears of the mean state contribute little to∇Q [Smith, 2007], but we retain them in our calculations for completeness.

4. The observed propagation speeds were calculated by Hughes from SSH observations in the following way. First, thin longitude (5 degrees)

and tall time (11.5 years) strips are bandpassed filtered in time from 5 to 57 weeks, then zonally averaged (at each time) and the annual and

semiannual cycles are removed. A Radon transform was then performed by shifting each longitude such that signals traveling at a speedc

line up horizontally, summing over longitude and taking the standard deviation in time. A “wavelikeness” parameter is also computed as the

peak value of the Radon transform divided by its mean. Based on advice from Hughes we have filtered out observations with wavelikeness

less than 1.5. Figure 2 shows global maps of the observed phase speed, wavelikeness (with black contour at 1.5), and a measure of wave

amplitude.

5. The correlation is defined as the absolute value of the real part of the pairwise linear correlation coefficient, which is similar to the coefficient

of the projection ofψm(z) ontoΦ1(z) except that the sum is not weighted by the layer thicknesses.



6. Suppose, instead of assuming that all the energy was in the first baroclinic mode, we imagined thatU(z) projected equally onto the barotropic

and first baroclinic mode. Then

ut =

»1

H

Zurms(z)

2dz

–1/2

=urms(0)

1 + Φ1(0)

»1

H

Z(1 + Φ1(z))2dz

–1/2

=

√2urms(0)

1 + Φ1(0),

since the modes are orthonormal. In the world ocean2 ≤ Φ1(0) ≤ 4, so the ratio of this projected value to one which is entirely first

baroclinic, as assumed in the text, is0.94 ≤√

2Φ1(0)/[1 + Φ1(0)] ≤ 1.13. An assumption of equipartition amongNz vertical modes

unambiguouslyreducesut, roughly by a factor of roughly√Nz .

7. Chelton provides eddy diameters, and here these are multiplied byπ to give wavelengths.

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−70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70−5

0

5

10

15

20

25

30

Wes

twar

d P

hase

spe

ed (

cm/s

)

Latitude

Figure 1. Westward phase speed estimated from Hughes’ data averaged from 170◦W to 120◦W (black

x’s) plotted against the standard linear, first baroclinic, long Rossby wave phase speed (solid line),

computed from theForget [2008] atlas.



Figure 2. Hughes’ analysis of surface altimetric data. Phase speed (top) “wavelikeness” (middle —

see text for details), with contour at 1.5 to differentiate regions that are wavelike and not wavelike, and

a measure of amplitude (bottom).



0 2 4−5

−4

−3

−2

−1

0

Dep

th (

km)

Φ1(z)

Figure 3. Map of first internal deformationradius(left), vertical structure of the first baroclinic mode

(right), Φ1(z), at the positions marked with colored x’s (at latitudes 60◦S, 45◦S, 30◦S, 15◦S, 0◦, 15◦N,

30◦N, and 45◦N, and longitude 150◦W). The lines are color-coded with dashed lines indicating the

northern hemisphere and solid lines the southern hemisphere.



Figure 4. Mean zonal velocityU (top), zonally averaged from 170◦W to 120◦W in the Pacific,

and meridional QGPV gradient (bottom) zonally averaged over the same region. The PV gradient is

normalized by the value of the planetary vorticity gradient,β, at 30 degrees. Note that the zero contour

is indicated by black contours and that the color axis is saturated.



−70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70−5

0

5

10

15

20

25

30

Wes

twar

d P

hase

spe

ed (

cm/s

)

Latitude

Figure 5. Hughes’ phase speed observations (black x’s) compared to linear theory in the presence of

a mean current: long-waves (gray solid line) and deformation scale waves (gray dashed line).



−70−60−50−40−30−20−10 0 10 20 30 40 50 60 70−5

0

5

10

15

20

25

30

Wes

twar

d ph

ase

spee

d (c

m/s

)

Latitude−70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 700

200

400

600

800

1000

1200

1400

1600

1800

2000

LatitudeW

avel

engt

h (k

m)

−70−60−50−40−30−20−10 0 10 20 30 40 50 60 70−5

0

5

10

15

20

25

30

Wes

twar

d ph

ase

spee

d (c

m/s

)

Latitude−70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 700

200

400

600

800

1000

1200

1400

1600

1800

2000

Latitude

Wav

elen

gth

(km

)

Figure 6. Top left: Phase speeds according to linear theory (solid gray line) adjusted to give the best

match to Hughes’ data (black x’s). The fit is done for a zonal average over 170◦W to 120◦W in the

Pacific. Top right: Fitted wavelengths at each latitude (black x’s, gray line is a smoothed version) along

with the deformation scale (thin solid line). Bottom panels: As in the top panels but zonally averaged

across all oceans.



−70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 70−5

0

5

10

15

20

25

30

Wes

twar

d ph

ase

spee

d (c

m/s

)

Latitude

Figure 7. The effects ofβ and mean currents on the prediction of phase speed from linear theory.

The thick dashed line corresponds phase speeds computed withU = 0 and the thin dash-dotted line

corresponds to those computed withβ = 0. In both cases the best-fit horizontal scale is used, that which

results in the phase speed plotted with the thick gray solid line in the presence of bothβ and mean

currents. (The x’s and thick gray solid line are identical to those in the top left panel of Figure 6 — all

data here are computed from a zonal average over the Pacific region 170◦W to 120◦W.)



1000 500 333 2500

0.05

0.1

0.15

Wavelength (km)

Inve

rse

times

cale

ω

(da

y−1)

U10k

U5k

ω(10°S)

ω(20°S)

ω(30°S)

Figure 8. Dispersion relations for fitted phase speeds as a function of zonal wavenumberk (with ` = 0)

for latitudes in the South Pacific (10◦S, 20◦S and30◦S), compared withωt = kut with two values of

ut: 5 and 10 cm s−1 (dashed lines).



0 10 20 30 40 50

−60

−40

−20

0

20

40

60

Latit

ude

|Urms

| at surface (cm/s)

Figure 9. Root mean square eddying surface velocities (left) from N. Maximenko’s drifter data, and

zonal average thereof (right).



−60 −40 −20 0 20 40 60

0

10

20

30

Velo

city

(cm

/s)

−60 −40 −20 0 20 40 600.1

1

10

Latitude

c R / u t

Figure 10. Top: Doppler shifted long-wave phase speed (thin black line), versus the root mean square

of the eddy velocityut (thick gray line) from Maximenko’s drifter data. It has been assumed that the

eddy velocity is entirely in the first baroclinic mode. Bottom: The ratiocR/ut with dashed curves at

ratios1/2 and2.



−70 −60 −50 −40 −30 −20 −10 0 10 20 30 40 50 60 700

100

200

300

400

500

600

700

800

900

1000

1100

1200

Latitude

Wav

elen

gth

(km

)

Figure 11. Comparison of fitted wavelengths over the global ocean (gray curve, taken from the bottom-

right of Figure 6) with Eden’s observed (black o’s) and simulated (black x’s) wavelengths, Chelton’s

observed wavelengths (black circles with solid line) and the deformation wavelength (thin black line).


interpretation of the propagation of surface altimetric...

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