A study of the dissipation of eddy kinetic energy and tracer dispersion in a submesoscale eddy field using subgrid mixing parameterizations Sonaljit Mukherjee 1 , Sanjiv Ramachandran 1 , Amit Tandon 1 , Amala Mahadevan 3 University of Massachusetts Dartmouth 1 ,Woods Hole Oceanographic Institution 2 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. . 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). . 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(1m 2 /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? 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 B y Salinity intrusions Isosurface, 36.6 PSU salinity transect at 7th inertial period Initialized fields PSU 0 C PSU kg/m 3 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 6 o C 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 Ageo. shear prod. Interscale transfer Buoyancy prod. Horiz. press. tran. Vert. press. tran. Geo. shear prod. Advection Sum 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-N B s -2 ×10 -4 0 1 2 3 4 z (m) -400 -200 0 N 2 Cycles/km 10 -2 10 -1 10 0 10 1 Π(κ)×4π 2 κ 2 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 KEPS, EKE, Along-front Cycles/km 10 -2 10 -1 10 0 10 1 Π(κ)×4π 2 κ 2 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 KEPS, EKE, Cross-front 10 -2 10 -1 10 0 10 1 S(κ)×4π 2 κ 2 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 KEPS, S', Along-front Cycles/km 10 -2 10 -1 10 0 10 1 S(κ)×4π 2 κ 2 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 KEPS, S', Cross-front Cycles/km 10 -2 10 -1 10 0 10 1 Π(κ)×4π 2 κ 2 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 Total and Ageo. EKE, σ θ =25.6 kg/m 3 a) b) d) e) c) -2 25.6 kg/m 3 25.7 kg/m 3 25.8 kg/m 3 25.9 kg/m 3 26.0 kg/m 3 26.1 kg/m 3 -2 25.6 kg/m 3 25.7 kg/m 3 25.8 kg/m 3 25.9 kg/m 3 26.0 kg/m 3 26.1 kg/m 3 -2 25.6 kg/m 3 25.7 kg/m 3 25.8 kg/m 3 25.9 kg/m 3 26.0 kg/m 3 26.1 kg/m 3 -2 25.6 kg/m 3 25.7 kg/m 3 25.8 kg/m 3 25.9 kg/m 3 26.0 kg/m 3 26.1 kg/m 3 -2 -1 Along-front, Total EKE Along-front, Ageo. EKE Cross-front, Total EKE Cross-front, Ageo. EKE Cycles/km -1 0 1/3 -1 0 1/3 -1 0 1/3 -1 0 1/3 -1 0 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 ˆ (u Õ i u Õ i ) ˆ t ¸ ˚˙ ˝ ˙ EKE = A ≠u j ˆ ˆ x j (u Õ i u Õ i ) B ¸ ˚˙ ˝ advection + ≠ Q a (u Õ i u Õ j ) A ˆ u i ˆ x j B geo R b ≠ Q a (u Õ i u Õ j ) A ˆ u i ˆ x j B ageo R b ¸ ˚˙ ˝ geo. shear production(P gr ) and ageo. shear production(P ar ) A B + (B Õ u Õ i ) i=3 ¸ ˚˙ ˝ buoyancy production B r ≠ 1 fl 0 ˆ ˆ x i (p Õ u Õ i ) ¸ ˚˙ ˝ pressure transport + A · ij ˆ u i ˆ x j B ¸ ˚˙ ˝ interscale transfer (‘ I ) ˆ ˆ t k = ˆ ˆ x i A ‹ m ‡ k ˆ ˆ x i k B i=3 ¸ ˚˙ ˝ downgradient transfer D k ≠ A u i ˆ ˆ x i k B i=1,2 ¸ ˚˙ ˝ Horizontal advection A h + A ≠u i ˆ ˆ x i k B i=3 ¸ ˚˙ ˝ Vertical advection A v + A ≠· ij ˆ u i ˆ x j B i=1,2;j =3 ¸ ˚˙ ˝ shear production P s =‹ m S 2 + 1 · B i 2 i=3 ¸ ˚˙ ˝ buoyancy production B s =≠‹ s N 2 ≠ ‘ ¸˚˙˝ subgrid dissipation , (3.17