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A study of the dissipation of eddy kinetic energy and tracer dispersion in a submesoscale eddy field using

subgrid mixing parameterizations

Sonaljit Mukherjee1, Sanjiv Ramachandran1, Amit Tandon1, Amala Mahadevan3

University of Massachusetts Dartmouth1,Woods Hole Oceanographic Institution2

High-resolution process studies for the Bay of Bengal

Acknowledgement We acknowledge support from the Office of Naval Research (N00014-09-1-0916, N00014-12-1-0101) and the National Science Foundation (OCE-0928138). We also acknowledge computational support from the Massachusetts Green High Performance Computing Cluster.

•  Large, W. G., J. C. McWilliams, and S. C. Doney, Oceanic vertical mixing: a review and a model with nonlocal boundary layer parameterisation, Rev. Geophys., 32, 363–403, 1994.

•  Mahadevan, A., A. Tandon, and R. Ferrari, Rapid changes in mixed layer stratification driven by submesoscale instabilities and winds, J. Geophys. Res., 115, C03017, doi:10.1029/2008JC005203, 2010.

Introduction ●  Submesoscale processes arise near fronts and play an

important role in vertical transport of nutrients within the mixed-layer as well as transferring energy to smaller scales.

●  Such flows are characterized by O(1) Rossby numbers and O(1) Richardson numbers.

●  3-dimensional Ocean model simulations at fine resolutions of O(100m to 1km) have revealed such flows, accompanied with intense vertical velocities of O(100m/day).

●  At resolved grid scales, such flows show a dominant balance between ageostrophic shear and dissipation of eddy kinetic energy.

●  The dynamics of turbulent fluxes in subgrid scales needs to be explored.

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Production and destruction of EKE in submesoscale simulations

Introduction ●  Submesoscale frontal processes play an important role in vertical

transport of nutrients within the mixed-layer and in transferring energy to O(10m - 100m) scales.

●  Such processes are characterized by O(1) Rossby numbers and O(1) Richardson numbers.

●  Three-dimensional ocean model simulations at fine resolutions of O(100m to 1km) have resolved such processes accompanied with intense vertical velocities of O(100m/day).

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References •  Fox-Kemper, B., R. Ferrari., and R. Hallberg, Parameterization of Mixed Layer Eddies. Part II: Prognosis and Impact. J. Phys. Oceanogr., 38,

1166–1179, 2008. •  Mahadevan, A, Modeling vertical motion at ocean fronts: Are nonhydrostatic effects relevant at submesoscales? Ocean Modelling 14 (2006) 222–

240. •  Kunze, E., Klymak, J. M., Lien, R.-C., Ferrari, R., Lee, C. M., Sundermeyer, M. A., and Goodman, L. (2015). Submesoscale water-mass spectra in

the sargasso sea. J. Phys. Oceanogr., 45(5):1325–1338.

Objective Previous numerical submesoscale simulations have typically implemented ad-hoc parameterizations for vertical diffusivities. ●  Study the spatial variability of subgrid dissipation in a

submesoscale eddy field. ●  Contrast the impact of subgrid eddy viscosity parameterizations

on resolved submesoscale flows and restratification. ●  Study the vertical structure of resolved and subgrid EKE budgets

using subgrid mixing parameterizations.

Initial condition showing the density front (white lines), with zonal velocity formed due to thermal wind balance Simulations done with PSOM

Initial mean vertical stratification (s-2) over the frontal region

Process modeling of dispersion by ageostrophic eddies below a

shallow mixed layer •  Flat gradient spectra of spice observed on isopycnal surfaces

below a shallow mixed layer during the Lateral Mixing Experiment (LatMix), in June 2011 in the Sargasso Sea.

•  O(1m2/s) diffusivity of tracers observed below the mixed-layer.

•  O(5km - 10km) long intrusions of salinity were observed below the mixed-layer. What is the underlying mechanism?

Velocity and density are in thermal-wind balance

Comparison of the simulated upper-ocean properties by

different 1D mixed-layer models

Advection by mixed-layer eddies at 7th inertial period form salinity intrusion at sub-surface depths, similar to the ones observed during LATMIX 2011 (see below).

Observed salinity transects from LATMIX 2011 (personal comm. with Craig Lee, APL Washington) showing

intrusions below the mixed-layer.

zonal velocity m/s

Isopycnal lines

T/S diagram obtained from LATMIX 2011. Red lines are observed profiles, and blue lines are from the idealized

domain.

Lateral buoyancy gradient By

Salinity intrusions

Isosurface, 36.6 PSU salinity transect at 7th inertial period

Initialized fields

PSU

0C

PSU

kg/m3

Intrusions

Intrusions

z (m

) z

(m)

W - E (km)

Initialized domain density lines

Enhanced dissipation in localized regions on the periphery of the eddies

Ageostrophic shear changes direction clockwise on the edge of the eddy due to non-linear Ekman advection by cyclonic relative vorticity. This deflection strengthens the total shear production on one side of the eddy and weakens the shear production on the other side.

CONST

KEPS

KPP

ML shallows more rapidly in KEPS

Isopycnal slumping

Vertical mixing

(Rudnick and Martin, 2002)

Continuous isopycnal slumping and vertical mixing reduces the lateral buoyancy gradients, thus reducing the APE. Stronger eddy diffusivities thus reduce the rate of restratification.

Price, Weller and Pinkel (PWP) (Price et al, 1986) •  Bulk mixed-layer model that implements convective

adjustment and a crude parameterization for shear instability at the mixed-layer base.

K-Profile Parameterization (KPP) (Large et al, 1994) •  Calculates the surface boundary layer, and evaluates

a cubic polynomial function as an approximation for the turbulent length scale to estimate eddy viscosities.

k-ε (Rodi, 1976) •  Implements two time-evolving equations for subgrid

Eddy Kinetic Energy (EKE) and dissipation rate ε. •  Estimates eddy viscosities and diffusivities

separately based on the local stratification and shear.

Surface Waves Processes Program (SWAPP) •  No near-inertial shear within the mixed layer. •  Intense near-inertial shear below the mixed layer.

Marine Light-Mixed Layer Experiment (MLML) •  22 day mixing phase followed by a 2½ month

restratification phase. •  SST elevates by 6oC from the mixing to the

restratification phase. SST amplitude Net SST

increment

•  Diurnal amplitude largest for KPP, followed by k-ε and PWP. •  SST increment largest for PWP at the end of the diurnal cycle,

followed by nearly equal increments by KPP and k-ε. •  The net SST increment at the end of a diurnal cycle accumulates

over multiple diurnal cycles, forming a large SST bias between thet PWP, and the KPP and k-ε models.

Leading order balance between dissipation and subgrid shear production

m2s-3 ×10-6-3 -1 1 3

z (m

)

-30-25-20-15-10

-5CONST2

m2s-3 ×10-6-3 -1 1 3

z (m

)

-30-25-20-15-10

-5KPP

m2s-3 ×10-6-3 -1 1 3

z (m

)

-30-25-20-15-10

-5

KEPS

m2s-3 ×10-7-2 -1 0 1 2

z (m

)

-100

-80

-60

-40

-20CONST2

m2s-3 ×10-7-2 -1 0 1 2

z (m

)

-100

-80

-60

-40

-20KPP

m2s-3 ×10-7-2 -1 0 1 2

z (m

)

-100-90-80-70-60-50-40-30-20

KEPS

Ageo. shear prod.Interscale transferBuoyancy prod.Horiz. press. tran.Vert. press. tran.Geo. shear prod.AdvectionSum

×10-7-2 3

-30

-20

-10a) b) c)

d) e) f)

Shear-driven layer near the surface, overlying a

buoyancy-driven layer

Subgrid EKE budget

Resolved EKE budget

Inertial and diurnal maxima

ε at 10m depth, after 10 inertial periods

Variability of SST with depth-integrated heat content

S-N (km)0 50 100 150

s-2

×10-8

-10

-5

0∇S-NB

s-2 ×10-40 1 2 3 4

z (m

)

-400

-200

0N2

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, EKE, Along-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, EKE, Cross-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

10-2 10-1 100 101

S(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, S', Along-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

Cycles/km10-2 10-1 100 101

S(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, S', Cross-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3

Total and Ageo. EKE, σθ=25.6 kg/m3

Along-front, Total EKEAlong-front, Ageo. EKECross-front, Total EKECross-front, Ageo. EKE

a) b)

d) e)

c)

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, EKE, Along-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, EKE, Along-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, EKE, Along-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, EKE, Along-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3

Total and Ageo. EKE, σθ=25.6 kg/m3

Along-front, Total EKEAlong-front, Ageo. EKECross-front, Total EKECross-front, Ageo. EKE

Cycles/km10-2 10-1 100 101

Π(κ

)×4π

2 κ2

10-1010-910-810-710-610-510-410-3 KEPS, EKE, Along-front

25.6 kg/m3

25.7 kg/m3

25.8 kg/m3

25.9 kg/m3

26.0 kg/m3

26.1 kg/m3

-10

1/3

-10

1/3

-10

1/3

-10

1/3

-10

1/3

•  Salinity spectra flatter than the EKE spectra, implying that salinity is stirred by ageostrophic eddies in the submesoscale range.

•  Variance reduces with depth, velocity gradient spectral slope close to -1.

•  Cross-front spectra flatter than along-front spectra.

•  Ageostrophic EKE spectra is flatter than the total EKE spectra.

Salinity and velocity gradient spectra on Isopycnal surfaces below the ML

While it is expected for the mixing and dissipation to be enhanced during convective

instability, our simulations show weaker dissipation at the destratifying edge and stronger

dissipation at the restratifying edge. This is because of the parameterization of the subgrid

mixing models which result in the dissipation to be in leading order balance with the subgrid

shear production. The subgrid EKE budget from the KEPS simulation further shows that

the subgrid buoyancy production is an order of magnitude less than the shear production.

Since the parameterized ‘ in the subgrid mixing models is proportional to the shear

production, the destratifying edge exhibits weak dissipation despite convective instability.

3.5 EKE budgets at resolved and subgrid scales

In this section we study the influence of different vertical mixing parameterizations on

the spatially averaged EKE budgets at resolved and subgrid scales, where the averaging is

done over the eddying region. Since the averaged budgets in the simulations CONST2 and

CONST1 are similar, we present only the results from CONST2, KEPS and KPP.

3.5.1 Resolved EKE budget

The following equation represents the different terms of the resolved-scale EKE budget:

ˆ(uÕiu

Õi)

ˆt¸ ˚˙ ˝˙EKE

=A

≠ujˆ

ˆxj

(uÕiu

Õi)

B

¸ ˚˙ ˝advection

+ ≠Q

a(uÕiu

Õj)

Aˆui

ˆxj

B

geo

R

b ≠Q

a(uÕiu

Õj)

Aˆui

ˆxj

B

ageo

R

b

¸ ˚˙ ˝geo. shear production(Pgr) and ageo. shear production(Par)

+ (BÕuÕi)i=3¸ ˚˙ ˝

buoyancy production Br

≠ 1fl0

ˆ

ˆxi

(pÕuÕi)

¸ ˚˙ ˝pressure transport

+A

·ijˆui

ˆxj

B

¸ ˚˙ ˝interscale transfer (‘I)

, (3.16)

65

While it is expected for the mixing and dissipation to be enhanced during convective

instability, our simulations show weaker dissipation at the destratifying edge and stronger

dissipation at the restratifying edge. This is because of the parameterization of the subgrid

mixing models which result in the dissipation to be in leading order balance with the subgrid

shear production. The subgrid EKE budget from the KEPS simulation further shows that

the subgrid buoyancy production is an order of magnitude less than the shear production.

Since the parameterized ‘ in the subgrid mixing models is proportional to the shear

production, the destratifying edge exhibits weak dissipation despite convective instability.

3.5 EKE budgets at resolved and subgrid scales

In this section we study the influence of different vertical mixing parameterizations on

the spatially averaged EKE budgets at resolved and subgrid scales, where the averaging is

done over the eddying region. Since the averaged budgets in the simulations CONST2 and

CONST1 are similar, we present only the results from CONST2, KEPS and KPP.

3.5.1 Resolved EKE budget

The following equation represents the different terms of the resolved-scale EKE budget:

ˆ(uÕiu

Õi)

ˆt¸ ˚˙ ˝˙EKE

=A

≠ujˆ

ˆxj

(uÕiu

Õi)

B

¸ ˚˙ ˝advection

+ ≠Q

a(uÕiu

Õj)

Aˆui

ˆxj

B

geo

R

b ≠Q

a(uÕiu

Õj)

Aˆui

ˆxj

B

ageo

R

b

¸ ˚˙ ˝geo. shear production(Pgr) and ageo. shear production(Par)

+ (BÕuÕi)i=3¸ ˚˙ ˝

buoyancy production Br

≠ 1fl0

ˆ

ˆxi

(pÕuÕi)

¸ ˚˙ ˝pressure transport

+A

·ijˆui

ˆxj

B

¸ ˚˙ ˝interscale transfer (‘I)

, (3.16)

65

3.5.2 Subgrid EKE budget

Among the different subgrid mixing schemes considered in this study, only the k ≠ ‘

scheme allows us to explore the subgrid EKE budget since it has a transport equation for the

parameterized subgrid EKE (k). The terms governing the evolution of k are shown below:

ˆ

ˆtk = ˆ

ˆxi

A‹m

‡k

ˆ

ˆxi

k

B

i=3¸ ˚˙ ˝downgradient transfer Dk

≠A

uiˆ

ˆxi

k

B

i=1,2¸ ˚˙ ˝Horizontal advection Ah

+A

≠uiˆ

ˆxi

k

B

i=3¸ ˚˙ ˝Vertical advection Av

+A

≠·ijˆui

ˆxj

B

i=1,2;j=3¸ ˚˙ ˝shear production Ps=‹mS2

+1·B

i

2

i=3¸ ˚˙ ˝buoyancy production Bs=≠‹sN2

≠ ‘¸˚˙˝subgrid dissipation

,

(3.17)

where ui is the resolved velocity and ·Bi is the subgrid buoyancy production. The terms Ah

and Av are the horizontal and vertical advection of k by the resolved-scale velocities. The

term Ps denotes the production of k at subgrid scales through the contraction of the subgrid

stress and the resolved-scale shear. Note that Ps is identical in magnitude but opposite in sign

to the interscale transfer term ‘I (equation 3.16), the sink in the resolved-scale EKE budget.

The term Bs is a downgradient parameterization for the subgrid buoyancy flux (Burchard

et al., 1999; Rodi, 1976). The term ‘ denotes the dissipation of EKE at the smallest scales,

which is parameterized in KEPS through a separate equation (3.5). The terms Ps and Bs

are parameterized based on the resolved shear and stratification respectively, and can be

obtained in the other subgrid mixing parameterizations as well.

70

3.5.2 Subgrid EKE budget

Among the different subgrid mixing schemes considered in this study, only the k ≠ ‘

scheme allows us to explore the subgrid EKE budget since it has a transport equation for the

parameterized subgrid EKE (k). The terms governing the evolution of k are shown below:

ˆ

ˆtk = ˆ

ˆxi

A‹m

‡k

ˆ

ˆxi

k

B

i=3¸ ˚˙ ˝downgradient transfer Dk

≠A

uiˆ

ˆxi

k

B

i=1,2¸ ˚˙ ˝Horizontal advection Ah

+A

≠uiˆ

ˆxi

k

B

i=3¸ ˚˙ ˝Vertical advection Av

+A

≠·ijˆui

ˆxj

B

i=1,2;j=3¸ ˚˙ ˝shear production Ps=‹mS2

+1·B

i

2

i=3¸ ˚˙ ˝buoyancy production Bs=≠‹sN2

≠ ‘¸˚˙˝subgrid dissipation

,

(3.17)

where ui is the resolved velocity and ·Bi is the subgrid buoyancy production. The terms Ah

and Av are the horizontal and vertical advection of k by the resolved-scale velocities. The

term Ps denotes the production of k at subgrid scales through the contraction of the subgrid

stress and the resolved-scale shear. Note that Ps is identical in magnitude but opposite in sign

to the interscale transfer term ‘I (equation 3.16), the sink in the resolved-scale EKE budget.

The term Bs is a downgradient parameterization for the subgrid buoyancy flux (Burchard

et al., 1999; Rodi, 1976). The term ‘ denotes the dissipation of EKE at the smallest scales,

which is parameterized in KEPS through a separate equation (3.5). The terms Ps and Bs

are parameterized based on the resolved shear and stratification respectively, and can be

obtained in the other subgrid mixing parameterizations as well.

70