1. Representing Gravity Current Entrainment in Large-Scale Ocean Models Robert Hallberg (NOAA/GFDL & Princeton U.) With significant contributions from Laura Jackson, Sonya Legg, and the Gravity Current Entrainment Climate Process Team: NOAA/GFDL:S. Griffies, R. Hallberg, S. Legg, L. Jackson* NCAR:G. Danabasoglu, P. Gent, W. Large, W. Wu* U. Miami:E. Chassingnet, T. Ozgokmen, H. Peters, Y. Chang* WHOI:J. Price, J. Yang, U. Riemenschneider* Lamont Doherty:A. Gordon George Mason:P. Schopf Princeton U.:T. Ezer (Plus ~12 active collaborators) *Postdocs funded by the CPT http://www.cpt-gce.org

2. An Idealized Rotating Overflow DOME Test Case 1 (Legg, et al., Ocean Modelling, 2006) Near-bottom tracer concentration with contours of buoyancy  x=500m,  z=30m MITgcm Simulation Tracer concentration just west of the inflow

3. An Idealized Rotating Overflow DOME Test Case 1 (Legg, et al., Ocean Modelling, 2006) Tracer concentration just west of the inflow  x=500m,  z=30m MITgcm Simulation Near-bottom tracer concentration with contours of buoyancy

4. Shear instability & entrainment Detrainment Geostrophic eddies x z y Downslope descent Bottom friction Physical processes in overflows Important Processes in Overflows Resolvable by large-scale models 1.Hydraulic control at sill 2.Geostrophic adjustment of plume along slope 3.Downslope transport of dense water (some model types?) 4.Some geostrophic eddy effects? 5.Detrainment at neutral density Require Parameterization 1. Exchange through subgridscale straits 2. Shear instability and entrainment (TURBULENCE!!!) 3. Bottom boundary layer mixing and drag processes (TURBULENCE!!!) 4. Some eddy effects? 5. Flow down narrow channels? Hydraulic control at sill Bottom-stress mixing

5. Overview A tour of overflows  Oceanic Gravity Currents are important in the formation and transformation of the majority of deep water masses. Important Processes in Typical Oceanic Dense Gravity Currents:  Hydraulic or tidal control of source water flows, often in narrow straits  Downslope descent (gravitational, Ekman driven, and eddy induced)  Shear-driven mixing at the plume top  Bottom boundary layer mechanical stirring within the plume  Thermobaric influences of the ocean’s nonlinear equation of state  Detrainment at the neutral depth Challenges for representing overflows in large-scale models:  Avoiding inherent problems with excessive numerical entrainment  Source water supply (representing the unresolved)  Studies of equilibrium stratified shear instability.  A new shear-driven turbulence mixing parameterization  A new bottom-turbulence mixing parameterization

8. Mediterranean Outflow Plume Without its 3-fold entrainment, Mediterranean Outflow water would fill the bottom of the Atlantic Gibraltar itself exhibits rectified tidal exchange in conjunction with hydraulic control Because of thermobaricity, salty Mediterranean water has a greater density at lower pressures, contributing to shallow detrainment. Gibraltar Velocities over the Tidal Cycle (CANIGO cruises Send & Baschek, JGR 2001)

9. Climatological Salinity at 1000 m Depth

12. Faroe Bank Channel and Denmark Strait Outflows Density along axis of Faroe Bank Channel Denmark Strait: J. Girton; FBC: C. Mauritzen, J. Price Denmark Strait Sea Surface Temperature

14. Abyssal Overflows – the Romanche Fracture Zone Potential Temperature along Romanche Fracture Zone Ferron et al., JPO 1998 Potential Temperature at 5000 m Depth

15. Shear instability & entrainment Detrainment Geostrophic eddies x z y Downslope descent Bottom friction Physical processes in overflows Steps in Adequately Representing Gravity Currents 1.Supply source water to the plume with the right rate and properties. 2.Model must be able to represent downslope flow without excessive numerical entrainment. 3.Parameterize entrainment & mixing to the right extent. 4.Parameterize subgridscale circulations? (e.g. eddies, flow in small channels). Hydraulic control at sill Bottom-stress mixing

16. Source water supply Source Water properties depend on the right large-scale circulation and properties. Several important source waters enter through very narrow channels!  Gibraltar is ~12 km wide.  Red Sea outflow channel is ~5 km wide.  Faroe Bank channel is ~15 km wide at depths that matter. Channels that are much smaller than the model grid require special treatment – e.g. partial barriers. The topography around Gibraltar, with a 1° grid (black), and the coastline (blue) that GFDL’s 1° global isopycnal model uses.

17. Representing Straits with Partially Open Faces (Work with A. Adcroft, GFDL) Partially open faces can dramatically improve simulations of overflows that pass through narrow straits. The model equations need to be modified to be energetically consistent. E.g. Sadourny’s 1975 Energy conserving discretization of the shallow water equations: Terms underlined in red are affected directly by using the partially open faces. Terms underlined in blue are affected indirectly (i.e. no code changes).             uvf hA A q j q y i q x ji h ji h  11,,

18. Resolution requirements for avoiding numerical entrainment in descending gravity currents. Z-coordinate: Require that AND to avoid numerical entrainment. (Winton, et al., JPO 1998) Suggested solutions for Z-coordinate models:  "Plumbing" parameterization of downslope flow: Beckman & Doscher (JPO 1997), Campin & Goose (Tellus 1999).  Adding a separate, resolved, terrain-following boundary layer: Gnanadesikan (~1998), Killworth & Edwards (JPO 1999), Song & Chao (JAOT 2000).  Add a nested high-resolution model in key locations?  No existing scheme is entirely satisfactory! Sigma-coordinate: Avoiding entrainment requires that Isopycnal-coordinate: Numerical entrainment is not an issue - BUT If resolution is inadequate, no entrainment can occur. Need

19. Diapycnal Mixing Equations in Isopycnic Coordinates In isopycnic coordinates, diapycnal diffusion is nonlinear The discrete form leads to a coupled set of nonlinear differential equations These can be solved implicitly and iteratively, with an arbitrary distribution of diffusivities to avoid the impossible time-step limit ( Hallberg, MWR 2000 ) The work-diffusivity relationship is exact in density coordinates. Entrainment can also be parameterized directly, based upon resolved shear Richardson numbers and a reinterpretation of the Ellis & Turner (1959) bulk Richardson number parameterization ( Hallberg, MWR 2000 ). This parameterization gives entraining gravity currents that are qualitatively similar to observations, but has subsequently been improved upon.

20. Constant Diffusivity Richardson Number Mixing

21. DOME Model Intercomparisons and Resolution Dependence (Legg et al., Ocean Modelling 2006) 2.5 km x 60 m10 km x 144 m50 km x 144 m 10 km x 25 Layer50 km x 25 Layer MITgcm (Z-coordinate) with Convective Adjustment HIM (isopycnal coordinate) with shear Ri# param.

22. Plume Entrainment as a Function of Resolution for 6 DOME Test Cases Final Plume Buoyancy (m s -2 ) Entrainment Rate Near Source (nondim.) Horizontal Grid Spacing (km) Solid lines: MITgcm (Z-coordinate) Dashed lines: HIM (Isopycnal coordinate) For full details, see Legg et al., Ocean Modelling 2006.

23. Parameterizing Overflow Entrainment: Observations of Bulk Entrainment in Oceanic and Laboratory Gravity Currents ( J. Price ) A bulk entrainment law applies, provided the Reynolds number is not quite small.

24. Examples of Gravity Current Mixing Parameterizations: Generic shear parameterizations – e.g. KPP (Large et al., 1994): Typically calibrated for the Equatorial Undercurrent. Two-equation turbulence closures (e.g. Mellor-Yamada; k-  ;  ). Plume-specific parameterizations – e.g. Ellison & Turner (1959) bulk Ri# parameterization reinterpreted for shear Ri# (Hallberg, 2000): This can be cast as a diffusivity,  is over an unstable region: May Need Resolution Dependence!

25. Simulated Mediterranean Outflow Plume (Papadakis et al., Ocean Modelling 2003) Zonal Velocity Salinity in 3 Isopycnal Layers Salinity

26. A Non-rotating Overflow Entering a Stratified Environment (Courtesy T. Özgökmen)

27. LES and Parameterized Overflow Entrainment (Xu, Chang, Peters, Özgökmen, and Chassignet, Ocean Modelling in press)

28. Failure and Success of Existing Parameterizations A universal parameterization can have no dimensional “constants”.  KPP’s interior shear mixing (Large et al., 1994) and Pacanowski and Philander (1982) both use dimensional diffusivities. The same parameterization should work for all significant shear-mixing.  In GFDL’s HIM-based coupled model, Hallberg (2000) gives too much mixing in the Pacific Equatorial Undercurrent or too little in the plumes with the same settings. To be affordable in climate models, must accommodate time steps of hours.  Longer than the evolution of turbulence.  Longer than the timescale for turbulence to alter its environment. 2-equation (e.g. Mellor-Yamada, k- , or k-  ) closure models may be adequate.  The TKE equations are well-understood, but the second equation (length-scale, or dissipation rate, or vorticity) tend to be ad-hoc (but fitted to observations)  Need to solve the vertical columns implicitly in time for: 1. TKE 2. Dissipation/vorticity 3. Stratification (T & S) 4. (and 5.) Shear (u & v)  Simpler sets of equations may be preferable.  Many use boundary-layer length scales (e.g. Mellor-Yamada) and are not obvious appropriate for interior shear instability. However, sensible results are often obtained by any scheme that mixes rapidly until the Richardson number exceeds some critical value.

29. 3-DNS of Shear Instability (L. Jackson, R. Hallberg, & S. Legg in prep.) Kelvin-Helmholtz instability 3D stratified turbulence z x Temperature (°C) z x Temperature during initial development of Kelvin-Helmholtz instabilities Representative instantaneous along-channel Cross-section in statistical steady state

30. Considerations for a Parameterization of Stratified Shear Instability S = ||∂U/∂z|| [s -1 ] Velocity shear N 2 = -g/  ∂  /∂z [s -2 ] Buoyancy Frequency H [m] Vertical extent of small Ri Q [m 2 s -2 ] Turbulent kinetic energy per unit mass u* = (  /  ) 1/2 [m s -1 ] Friction velocity (for boundary turbulence) z* [m] Distance from boundary (for boundary turbulence) Mixing should vanish if the shear Richardson number (Ri = N 2 /S 2 ) exceeds ~1/4 everywhere Vigorous mixing may extend past the region of small Ri. Homogeneous stratified turbulence is often characterized by the buoyancy length scale Kelvin-Helmholtz (K-H) saturation velocity scales are ~ H S.  K-H instabilities span the region of small Ri, i.e. length scales of ~ H.  Mixing-length arguments suggest peak K-H-type diffusivities scaling as ~ H 2 S. Near solid boundaries, length scales are proportional to the distance from the boundaries, and diffusivities are ~ 0.4u*z*.

31. The diffusion of density can be linked to entrainment parameterizations by combining the density conservation equation: with the continuity equation in density coordinates: The latter equality is ill-behaved when ∂  /∂ z=0, but with constant stratification it reduces to ET parameterisation (Hallberg, 2000) Translating “Entrainment Rate” parameterizations into diffusive parameterizations (L. Jackson)

32. Properties: Uses a length scale which is a combination of the width of the low Ri region (where F(Ri)>0) and the buoyancy length scale L Buoy = Q 1/2 /N. Decays exponentially away from low Ri region Vertically uniform, unbounded limit: Ellison and Turner limit (large Q) reduces to form similar to ET parameterisation Unstratified limit: similar to law-of-the-wall theories of parabolic diffusivity between two boundaries and log-like profiles of velocity near the boundaries. S=||U z ||  = 0 at solid boundaries Entrainment-law derived theory for Shear-driven mixing

33. Assumptions: Q reaches steady state faster than background flow is evolving so no DQ/Dt term Assume Pr = 1 (for now) Q 0 needed to avoid singularity in diffusivity equation (solution not sensitive to Q 0 and 0 ) Parameterization of dissipation as c(Q-Q 0 )N Q intended for use in diffusivity equation is due to turbulent kinetic energy only - difficult to compare to results from DNS because of internal waves. TKE Budget to Complement Proposed Diffusivity Equation

34. Equilibrium DNS of Shear-driven Stratified Turbulence Non-hydrostatic direct numerical simulations (MITgcm) 2m x 2m x 2.5m with grid size ~ 2.5mm in centre Molecular viscosity and diffusivity, Kolmogorov scale mostly resolved. Cyclic domain in x,y Shear and jet profiles Statistically steady state reached Force average velocity profiles to evolve to given profile Initially constant stratification and relaxed to initial density profile All profiles are spatially averaged in x and y and time averaged

35. 3-DNS of Shear Instability (L. Jackson, R. Hallberg, & S. Legg in prep.) Kelvin-Helmholtz instability 3D stratified turbulence z x Temperature (°C) z x Temperature during initial development of Kelvin-Helmholtz instabilities Representative instantaneous along-channel Cross-section in statistical steady state

36. DNS data New parameterisation (Jackson et al.) ET parameterisation (Hallberg 2000) F(Ri) = 0.15*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.9 F(Ri) = 0.15*(1-Ri/0.8)/(1+1.0*Ri/0.8), c=1.7 F(Ri) = 0.12*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.24 DNS Shear-Instability Results and the Proposed Parameterization Buoyancy flux (m 2 /s 3 )

37. DNS of Shear Instability and Existing 2-equation closures Jackson et al., proposed parameterization: Black: DNS Results Green: GOTM k  Blue: GOTM k  Red: Mellor-Yamata 2.5 Buoyancy flux (m 2 /s 3 )

38. DNS Jet results Buoyancy flux (m 2 /s 3 ) DNS data New parameterisation (Jackson et al.) ET parameterisation (Hallberg 2000) F(Ri) = 0.15*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.9 F(Ri) = 0.15*(1-Ri/0.8)/(1+1.0*Ri/0.8), c=1.7 F(Ri) = 0.12*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.24

39. Existing 2-equation Closures Compared to DNS Jet Black: DNS Results Green: GOTM  Blue: GOTM  Red: Mellor-Yamata 2.5 Jackson et al., proposed parameterizations: Buoyancy flux (m 2 /s 3 )

40. Diagnosed diapycnal diffusivity (m 2 s -1 ) Gradient Richardson number Diapycnal Diffusivities Diagnosed from 3-D DNS

41. Shear Instability with a Larger Ri# Not yet equilibrated? Buoyancy flux (m 2 /s 3 )

42. 500 m x 30 m MITgcm Ellison & Turner Mixing Only 10 km x 25 layer HIM At the start of the CPT, with thick, nonrotating plumes entering ambient stratification, GFDL’s Isopycnal coordinate model (HIM) would give plumes that split in two. Such split plumes do not occur in nonhydrostatic “truth” simulations. Illustrating the power of the CPT paradigm

43. Observed profiles from Red Sea plume from RedSOX (H. Peters) Well-mixed Bottom Boundary Layer Actively mixing Interfacial Layer Shear Ri# Param. Appropriate Here.

44. Bottom Boundary Layer Mixing Diapycnal mixing of density requires work. The rate at which bottom drag extracts energy from the resolved flow is straightforward to calculate. Assumptions:  20%? of the extracted energy is available to drive mixing.  Available work decays away from the bottom with e-folding scale of  Mixing completely homogenizes the near bottom water until the energy source is exhausted. Legg, Hallberg, & Girton, Ocean Modelling, 2006

45. 500 m x 30 m MITgcm Ellison & Turner + Drag MixingEllison & Turner Mixing Only 10 km x 25 layer HIM With thick plumes, both Interfacial and and Drag-induced Mixing are needed. (Legg et al., Ocean Modelling, 2006)

46. Double Mediterranean plumes without bottom-drag mixing Year 5 salinity along 38.5°N in GFDL’s 1° Global Isopycnal Model Adding the Legg et al. bottom-drag mixing parameterization leads to dramatic improvements in an IPCC-class ocean model.

47. Summary Overflows are critical in the formation of most deep-ocean water masses. Turbulent mixing with the right rate is critical for models to obtain the right properties. (Otherwise in a stratified ambient environment, the plunging plume entrains the wrong water.) Large-scale models require parameterizations of such mixing that capture both the equilibrium turbulence and (sometimes) its equilibrium modification of the resolved flow. Mature Kelvin-Helmholtz-like mixing is significant in the interfacial layers atop gravity currents.  Existing parameterizations do not appear to work very well in detail based on comparisons with DNS (although they may work well enough for some overflows).  Laura Jackson (Princeton/GFDL CPT postdoc) has a new 2-equation (diffusivity – TKE) shear-driven turbulent mixing parameterization that looks very promising. Bottom-stress driven turbulence is significant for homogenizing the bottom boundary layer, and must be parameterized. http://www.cpt-gce.org