potential vorticity transport in two-layer quasigeostrophic - - PowerPoint PPT Presentation

Role of coherent eddies in potential vorticity transport in two-layer quasigeostrophic turbulence.

Wenda Zhang1, Christopher L.P. Wolfe1, Ryan P. Abernathey2

1Stony Brook University 2Columbia University/LDEO

Introduction– two definitions of an “eddy”

Eulerian view:

• Eddies are fluctuations from mean state: 𝑣′ = 𝑣 − ത

𝑣;

• Transport by stirring the fluids: 𝑣′𝑑′;
• Local;
• Parameterization as a turbulent diffusion process.

Lagrangian view:

• Eddies are coherent structures;
• Transport by trapping the fluids;
• Nonlocal;
• Ocean mesoscale eddies (Chelton et al., 2011), Atmosphere polar

vortex (McIntyre 1995) Frenger et al. (2015) Anticyclone Cyclone

Transport by coherent eddies in ocean

Zhang et al 2014

Zhang et al. (2014) Abernathey et al. (2018)

Questions remained

• What flow regimes favor coherent eddies?
• How different are the dynamics and transport properties

between coherent eddies and background turbulent flows?

• Is the transport due to coherent eddies significant?

Method

• Two-layer quasi-geostrophic (QG) model to mimic the Southern Ocean (𝑀𝑒 = 15 𝑙𝑛)
• Double periodic boundary conditions.
• Horizontal resolution: 2.34 km (512 × 512)
• Three different frictional strengths following Wang et al. (2016):

𝑠

ek −1 = 10, 20, and 40 days

—referred to as the strong friction, control and weak friction cases, respectively.

Advection of Lagrangian particles

• The kinematic equation of the Lagrangian

particles: 𝑒𝒚 𝑒𝑢 = 𝒗 𝒚 𝑢 , 𝑢

• Particle spacing: half of the model grid spacing

(1.17 km)

• Particles were advected for 90 days in the upper

layer for each experiment.

• Positions, velocities, vorticities and PV on

particles were saved daily.

Particle trajectories in 30 days

Identification of coherent eddies

• Adopt the Lagrangian Averaged Vorticity Deviation (LAVD) technique of Haller et

al.(2016) to detect coherent eddies.

• The cores of coherent eddies are identified as the maxima of LAVD, defined as:

𝑀𝐵𝑊𝐸𝑢0

𝑢1 𝒚𝟏 =

1 𝑢1 − 𝑢0 න

𝑢0 𝑢1

𝜂 𝒚 𝒚𝟏, 𝑢 , 𝑢 − ҧ 𝜂 𝑒𝑢

Coherency index (CI CI): CI = 1 2 𝑆2 − max[𝜏2 𝑢𝑗 ] 1 2 𝑆2 𝜏2 𝑢𝑗 =< 𝑌 𝑢𝑗 −< 𝑌 𝑢𝑗 > 2 >, 𝑢𝑗 ∈ (𝑢0, 𝑢1) Threshold: CI > −0.75

30-day coherent eddy detection results

• Red curves: initial outer

boundary of the eddies

• Colored dots: final positions
• f particles inserted initially

in the red curves

• Black curves: trajectories of

particle center during 30 days’ drift.

Occurrence frequency and radius distribution of coherent eddies

• Short-lived eddies are much more than the longer-lived eddies.
• The number of eddies becomes fewer as the friction reduces.
• Average radius of the coherent eddy cores is close to the Rossby deformation radius

𝑀𝑒 (15 km) of the model.

Eddy meridional displacement

• Opposite meridional propagation preference (60%) between cyclones and

anticyclones.

• Due to Beta effect and nonlinear advection.
• A vortex has tendency to return to a rest latitude (Rossby 1949, McWilliams

and Flierl et al., 1979).

Cushman-Roisin (1994) What does this mean to meridional PV transport by coherent eddies?

Advective PV transport by coherent eddies

• The Lagrangian meridional PV flux due to the coherent eddies:

𝑅𝑑 =

𝐵𝑤′𝑟′ ҧ 𝐵

• A is a masking function which is 1 for particles inside coherent

eddies and 0 outside.

• The PV flux due to incoherent motions:

𝑅𝑗𝑜𝑑 = 1 − 𝐵 𝑤′𝑟′ 1 − 𝐵

• The coherent PV flux 𝑅𝑑 is systematically positive (upgradient).
• PV transport due to drift of coherent eddies is less than 10% of

total PV transport.

Advective PV transport by coherent eddies

• However, coherent eddies also induce

the flows in the far field.

• Piecewise PV inversion shows that the

meridional PV transport by flow induced by coherent eddies is 10-30% of the total PV transport and is systematically downgradient.

Conclusion

• Materially coherent eddies are prevalent in the flow regimes in this

study, with stronger friction associated with more coherent eddies.

• Meridional propagation preference of coherent eddies gives rise to

upgradient PV transport.

• The PV transport by trapping of coherent eddies is relatively small

due to the dynamical constraint, while the PV transport by the flow induced by the coherent eddies is more significant.

Supplement: Method

• Two-layer quasi-geostrophic (QG) model to mimic the Southern Ocean
• Forced dissipative PV evolution equations:

𝜖𝑟1 𝜖𝑢 + 𝑉1 𝜖𝑟1 𝜖𝑦 + 𝐾(𝜔1, 𝑟1) + [𝐺

1(𝑉1 − 𝑉2) + β] 𝜖𝜔1

𝜖𝑦 = ssd, 𝜖𝑟2 𝜖𝑢 + 𝑉2 𝜖𝑟2 𝜖𝑦 + 𝐾(𝜔2, 𝑟2) + [𝐺

2(𝑉2 − 𝑉1) + β] 𝜖𝜔2

𝜖𝑦 = −𝑠

ek∇2𝜔2 + ssd.

The potential vorticity are: 𝑟1 = 𝛼2𝜔1 + 𝐺

1 𝜔1 − 𝜔2

𝑟2 = 𝛼2𝜔2 + 𝐺

2 𝜔2 − 𝜔1 ,

• where

𝐺

1 = 𝑙𝑒

2

1+𝜀 , 𝐺 2 = 𝑙𝑒

2𝜀

1+𝜀 ,

• 𝑙𝑒 is the inverse of Rossby deformation radius 𝑀𝑒, and 𝜀 = 𝐼1/𝐼2 is the ratio of the thickness of two layers.
• This study uses the same parameters as Wang et al. (2016):

𝑀 = 1200 km, 𝑀𝑒 = 15 km, 𝐼1 = 800 m, 𝜀 = 0.25, 𝑉1 = 0.04 m s−1, 𝑉2 = 0, and 𝛾 = 1.3 × 10−11m−1s−1.

pyqg (Abernathey et al. 2015)

Supplement: Enstrophy equation analysis

• PV transport due to drift of coherent eddies is less than 10% of total PV transport.

𝜖 𝜖𝑢 1 2 𝑟′2 + ഥ 𝒘 ∙ ∇ 1 2 𝑟′2 = −𝒘′𝑟′ ∙ ∇ത 𝑟 + 𝐸′𝑟′ 0 ≈ 1 2 𝑒 𝑒𝑢 න q′2 𝑒𝑊 = − 𝜖ത 𝑟 𝜖𝑧 (න 𝑤𝑑𝑝ℎ

′

𝑟𝑑𝑝ℎ

′

𝑒𝑊 + න 𝑤𝑗𝑜𝑑

′

𝑟𝑗𝑜𝑑

′

𝑒𝑊) + න 𝐸′𝑟′𝑒𝑊