ReferencesCharney, J. G.: 1955, The gulf stream as an inertial boundary layer.Proceedings of the National Academy of Sciences,

41, 731–740.

Fofonoff, N. P.: 1954, Steady flow in a frictionless homogenous ocean.Journal of Marine Research, 13, 254–262.

Fox-Kemper, B.: 2003, Wind-driven barotropic gyre II: Effects of eddies and low interior viscosity.submitted to theJournal of Marine Research.

Fox-Kemper, B. and J. Pedlosky: 2003, Wind-driven barotropic gyre I: Circulation control by eddy vorticity fluxesto a region of enhanced removal.submitted to the Journal of Marine Research.

Greatbatch, R. J. and B. T. Nadiga: 1999, Four-gyre circulation in a barotropic model with double-gyre wind forcing.Journal of Physical Oceanography, 30, 1461–1471.

Griffa, A. and S. Castellari: 1991, Nonlinear general circulation of an ocean model driven by wind with a stochasticcomponent.Journal of Marine Research, 49, 53–73.

Holm, D. D. and B. T. Nadiga: 2003, Modeling mesoscale turbulence in the barotropic double gyre circulation.Journal of Physical Oceanography, 33, 2355–2365.

Munk, W. H.: 1950, On the wind-driven ocean circulation.Journal of Meteorology, 7, 79–93.

Ozgokmen, T. and E. P. Chassignet: 1998, Emergence of inertial gyres in a two-layer quasigeostrophic ocean model.Journal of Physical Oceanography, 28, 461–484.

Pedlosky, J.: 1965, A study of the time dependent ocean circulation.Journal of the Atmospheric Sciences, 22,267–272.

Stommel, H. M.: 1948, The westward intensification of wind-driven ocean currents.Transactions, American Geo-physical Union, 29, 202–206.

Sverdrup, H. U.: 1947, Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of theeastern Pacific.Proceedings of the National Academy of Sciences, 33, 318–326.

VI. Conclusions

• Basin modes dominate the variability of the basin interior.

• These basin modes resonate and are forced by instabilities of the western boundary current.

• In the basin interior, basin modes, not vortices or mean-mean interaction dominates the non-linearity.

• Calculating the nonlinear interaction of analytic basin modes with the variances from EOFsgives good agreement for deviations from Sverdrup (1947) interior flow.

a) The time-mean streamfunction minusthe Sverdrup (1947) interior solution forthe Reb= 0.25, Rei= 5 calculation revealsa counter-rotating and co-rotating gyrepattern irreconcilable with a local down-gradient absolute vorticity flux. b) Theprediction using basin modes (1,1), (1,2),(2,1), and (2,2) with variances from EOFdiagnosis.

a) The time-mean streamfunction ina elongated basin with Reb=3, Rei=3.There is wind forcing only in the westernhalf of the basin. b) The prediction withanalytic basin modes (3,1), (2,1), (1,2),(1,1), and (2,2) with variances from EOFdiagnosis.

V. Confirmation of TheoryThe variances of a few basin modes are deduced from the EOFs of a model calculation. Thenonlinear interaction of theanalytic basin modes produces good agreement with the interiorsolution, including the counter-rotating regions.

Counter-rotating gyres in other models have been attributed to Fofonoff (1954) gyres (predictedby inviscid statistical mechanics Griffa and Castellari, 1991;Ozgokmen and Chassignet, 1998),mixing of absolute vorticity (Greatbatch and Nadiga, 1999), and nonlinear dispersion (Holm andNadiga, 2003). However, the Fofonoff (1954) primary is mean vorticity advection andβ-term;some regions would require ’up-gradient’ absolute vorticity fluxes; and the gyres have a sensitivedependence on the frequency of the boundary current instabilities not captured by the dispersionmodel.

In summary,

• EOFs strongly resemble spatial structure of basin mode standing waves with two EOFs perbasin mode.

• PSDs of EOF presence have peaks at basin mode frequencies.

• Vorticity EOFs reveal that most variance in in boundary current region, but frequency matchesthat of the resonating modes.

• Thus, there are basin modes forced by boundary current instabilities.

• The counter-rotating gyres occur on the eastern side, where basin modes are the most importantvariability, and their vorticity balance depends critically on eddy flux divergences.

IV. TheoryConsider a grossly simplified model, replacing Laplacian friction with bottom drag and repre-senting the boundary mode instabilities as an additional forcing.

∂∇2ψ

∂t+∂ψ

∂x+ δ2

IJ(ψ,∇2ψ) + δS∇2ψ = − sin(πy) + Af sin(nfπy) cos(ωf t)e−x/δf . (5)

Consider the weakly nonlinear perturbation series:ψ = ψ0 + εψ1 + . . . under the assumption thatδI � 1. Theψ0 equation is linear, so it is uncoupled into a steady equation (similar to Stommel(1948)) and a time-dependent equation (similar to Pedlosky, 1965). By differentiating the linearsolutions, the first nonlinear correction can be calculated. If the periodic forcing is resonant witha basin mode, then the results are to lowest order inδS:

δ2IJ(ψ0,∇2ψ0) ≈ 1

2π3δSδ

2I sin(2πy), (6)

δ2IJ(ψ′0,∇2ψ′0) ≈ 2m3nfπ

4ω4f |ϕ0|2δ2

I

δ2S

sin(2mπx) sin(2nfπy)

(1− 2(−1)me

− 1δf cos

(1

2ωf

)+ e− 2δf

).

The resulting mean vorticity flux convergence is very large within the western boundary currentbutO(δSδ

2I) outside it. On the other hand, the eddy term isO(|ϕ0|2δ2

I/δ2S) outside of the forcing

region. The analytic eddy flux divergence gives approximate nonlinear interaction outside of theboundary current region, resulting in a correction to Sverdrup flow.

ψ0 + ψ1 = (1− x) sin(πy) +

∫ x

1δ2IJ(ψ′0,∇2ψ′0)dx. (7)

Although the strength of each basin mode depends in a complicated way on the boundary currentinstabilities, the eddy interaction as a function of the average variance of the basin mode, whichwe can deduce from the EOFs.

δ2IJ(ψ′0,∇2ψ′0) ≈ 4π4mnf (m2 + n2

f )δ2I sin(2mπx) sin(2nfπy)

∫ 1

0

∫ 1

0(ψ′0)2dxdy, (8)

(a) and (b) show the meridional and zonalaverages, respectively, of terms in (5) inthe region where ψ < 0 (the counter-rotating gyre) from the Rei=Reb=3 elon-gated basin calculation with wind only inthe western half of the basin.

(a) and (b) show the meridional and zonalaverages, respectively, of vorticity flux con-vergences in the region where ψ < 0 (thecounter-rotating gyre) from the Reb=0.25,Rei=5 calculation.

III.b. Resonating Basin Modes

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #1:% of var.=23.5

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #2:% of var.=18.3

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #3:% of var.=17.7

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #4:% of var.=13.3

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #5:% of var.=8.95

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #6:% of var.=3.47

x

y

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.018

freq.

#1 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.0111

freq.

#2 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.0111

freq.

#3 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.018

freq.

#4 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.000391

freq.

#5 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.0119

freq.

#6 P

SD

The ψ EOFs and PSDs of EOF presencefor a Reb=0.25, Rei=5 calculation.

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #1:% of var.=12.1

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #2:% of var.=9.21

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #3:% of var.=6.52

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #4:% of var.=5.55

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #5:% of var.=5.07

x

y

0 0.5 10

0.2

0.4

0.6

0.8

1EOF #6:% of var.=3.59

x

y

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.0119

freq.

#1 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.0119

freq.

#2 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.000391

freq.

#3 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.0111

freq.

#4 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.000391

freq.

#5 P

SD

10−3

10−2

10−1

10−4

10−2

100

102

freq. of max=0.0115

freq.

#6 P

SD

The ζ EOFs and PSDs of EOF presencefor a Reb=0.25, Rei=5 calculation.

0 1 20

0.5

1EOF #1:% of var.=42.4

x

y

0 1 20

0.5

1EOF #2:% of var.=41.6

x

y

0 1 20

0.5

1EOF #3:% of var.=3.3

x

y

0 1 20

0.5

1EOF #4:% of var.=3.26

x

y

0 1 20

0.5

1EOF #5:% of var.=2.94

x

y

0 1 20

0.5

1EOF #6:% of var.=2.86

x

y

10−3

10−2

10−1

10−10

10−8

10−6

10−4

10−2

100

102

freq. of max=0.0109

freq.

#1 P

SD

10−3

10−2

10−1

10−10

10−8

10−6

10−4

10−2

100

102

freq. of max=0.0109

freq.

#2 P

SD

10−3

10−2

10−1

10−10

10−8

10−6

10−4

10−2

100

102

freq. of max=0.0121

freq.

#3 P

SD

10−3

10−2

10−1

10−10

10−8

10−6

10−4

10−2

100

102

freq. of max=0.0121

freq.

#4 P

SD

10−3

10−2

10−1

10−10

10−8

10−6

10−4

10−2

100

102

freq. of max=0.0227

freq.

#5 P

SD

10−3

10−2

10−1

10−10

10−8

10−6

10−4

10−2

100

102

freq. of max=0.0234

freq.

#6 P

SD

The ψ EOFs and PSDs of EOF presencefor a Reb=1, Rei=1 calculation in an elon-gated basin.

III.a. Counter-Rotating Gyres

The enhanced viscosity near the boundary al-lows for much higher interior Reynolds numberwithout inertial domination (Fox-Kemper andPedlosky, 2003).

The time-mean streamfunction for a num-ber of parameter settings shows regionsrotating counter to the wind stress.

III. Results

I. AbstractRelatively high Reynolds number calculations of a barotropic ocean model reveal counter-rotating gyres on the eastern side of the basin. These unintuitive regions rotate in a directionopposite to the wind forcing direction, and are sometimes associated with strong up-gradient vor-ticity fluxes. An analysis of the empirical orthogonal functions of some numerical calculationsreveals that much of the variance is in oscillations resembling basin modes. A simple non-localmodel of the nonlinear interaction of forced-dissipative basin modes, together with the observedvariance of each mode in the numerical calculation, gives excellent agreement with the stream-function and dynamical balances of the counter-rotating gyres.

(Fox-Kemper, 2003)

II. ModelThe model results presented here are from a 257x257 Chebyshev polynomial pseudo-spectralnumerical barotropic model in a rectangular basin with spatially-variable viscosity to roughlyparameterize boundary physics not directly represented in the model. The nondimensional equa-tions governing the model are:

∂ζ∂t +∇ · (xψ + δ2

Iuζ − δ3M∇ζ + δS∇ψ) = − sin(πy), (1)

ζ = ∇2ψ, (2)

δ3M =

δ3I

Rei+(δ3I

Reb− δ3

IRei

)(e−x/δd + e−(1−x)/δd

), (3)

δd ≡ δI√Rei, (4)

where ψ (streamfunction) and ζ (relative vorticity) are determined during integration. Boundaryconditions are slip (ζ=0) on the ’fluid’ boundaries and no-slip (∂ψ∂x = 0) on the ’solid’ boundaries,as well as impermeability on all boundaries (ψ = 0). The basin domain is y between 0 and 1 andx between 0 and xe. The other parameters are δI (Charney, 1955, inertial boundary layer width),δS (Stommel, 1948, frictional boundary layer width), and Rei and Reb are Reynolds numbers forthe interior and boundary viscosity (Munk, 1950). Throughout, δI is 0.02 and δS is 0, while Reiand Reb vary.

Baylor Fox-KemperNOAA Climate and Global Change Program and Princeton University Atmospheric and Oceanic Sciences ProgramP.O. Box 308, Princeton, NJ 08542, baylor@princeton.edu

OS21F-12

Counter-Rotating Gyres as a Non-Local Effect of Resonating Basin Modes