Constraints for adequately representing boundary layers

1. An Overview of Numerical Representationsof Gravity Currents in Climate ModelsThe Modeling Center PerspectiveRobert Hallberg, Stephen GriffiesNOAA Geophysical Fluid Dynamics LaboratoryG. Danabasoglu, W.G. Large, W. WuNational Center for Atmospheric Research

2. Constraints for adequately representing boundary layers Z-coordinate: Require that AND • If resolution is inadequate, excessive non-physical entrainment occurs. Suggested solutions: • Unresolved specification of downslope flow: Beckman & Doscher (JPO 1997), Campin & Goose (Tellus 1999). • Adding a separate, resolved boundary layer: Gnanadesikan (unpub., ~1998), Killworth & Edwards (JPO 1999),Song & Chao (JAOT 2000). Sigma-coordinate: Require that • If resolution is inadequate, excessive, non-physical entrainment occurs. • Adequate representation can be achieved by resolving the BBL, with a straightforward implementation of a turbulence closure scheme. • Pressure gradient errors are a potential difficulty. Isopycnal-coordinate: Require that • If resolution is inadequate, no entrainment can occur. • With adequate resolution for mixing, numerical implementation is still tricky, requiring implicit solution of coupled, nonlinear equations, but these issues are addressed by Hallberg (2000). • Diapycnal mixing might be solved in depth-coordinates, with remapping back into density space.

3. Evaluation of Schemes - Truth The ultimate test of the success of an overflow scheme is its impact in global models at the resolutions (~1° now) used for coupled climate studies. • Both the mean state (circulation and watermasses) and their sensitivity to forcing changes are important. • Indirectly motivated changes (e.g. topography or resolution) must be evaluated. • This ultimate truth is very complicated and strongly dependent on many choices. Regional studies are of great value because they are more highly constrained. • Boundary conditions can ensure that the environment matches the real world. • Simulations can be directly compared with observations. • Higher resolution simulations are similarly valuable for assessing what is of primary importance, but with caveats about differences in topography, forcing, etc. Idealized studies provide the most promising avenue for comparison with converged “truth”. • High resolution simulations with identical forcing and domains will hopefully provide both truth and understanding. • Many potential differences between models can be avoided in idealized studies. • Parameter-space sensitivity studies are best carried out in this context. All three approaches (global, regional, and idealized) will play important and complementary roles in achieving the objectives of this CPT.

4. Beckmann and Döscher – Journal of Physical Oceanography 1997 2 parts: (1) A fixed fraction (g) of the tracer advection in the bottommost cell follows the bottom. Momentum equations are only solved in the Z-coordinate. (2) Diffusion of tracer occurs along the bottom when the bottom density gradient is unstable. Strengths: • This approach has demonstrated value when Dz/HBBL≈ 1. Concerns: • Coefficients require tuning – probably different values for different overflows. • Strong vertical resolution dependence – does not work well with partial cells. • Limited underlying physics – weak dependence on flow state. • Convective overturning can occur with multiple-level steps.

5. Campin and Goose – Tellus 1999 Argue that at coarse (3°) resolution, bottom flows are unresolvable. Hence, they suggest a downslope / along-isopycnal flow proportional to the density difference at the top of the slope. Tracers are advected to the neutral depth in the bottommost grid cell with a velocity: UDownslope = g g Dr/ro only when Dr=rShelf-rOffshore>0. The coefficient g=da/m, where 0<d<1 and m is a Rayleigh drag coefficient. • Strengths: • Demonstrated improvement in a global 3° model over simulations with no parameterization. • Does not drive deep convective overturns. • Philosophically self-consistent. • Concerns: • Empirical parameter g needs to be tuned for potentially thousands of locations; the theory to predict g is limited. • Has strong dependence on vertical resolution. • Limited underlying physics – weak dependence on flow state.

6. Gnanadesikan – Unpublished ~1998 Appends a fixed depth (~100 m) BBL to the bottom of a Z-coordinate OGCM. Simplified momentum equations are solved in the BBL (omitting momentum advection terms). • Strengths: • Able to get dense water to flow downslope. • Demonstrated usefulness in a global 1° model (Nakano 2000). • Independent of vertical resolution in interior model. • Works with partial cells and arbitrary bottom depth. • Concerns: • BBL entrainment and detrainment are due to BBL convergence and divergence; there is nonphysical mixing. • Pressure gradient errors drove huge upslope flow at equator. (Nakano simply turns the BBL off equatorward of 30°.) • As implemented in MOM3, was extremely pervasive in code - not modular. • Does not capture the extent and location of entrainment; it has been abandoned at GFDL in favor of other approaches.

7. Killworth & Edwards – Journal of Physical Oceanography 1999 Embeds a BBL with a diagnostically determined, spatially and temporally varying thickness inside of a Z-coordinate OGCM. Full momentum equations are solved in the slab-BBL. Thickness is limited by rotation and stratification: • Strengths: • Has potentially realistic dependence of BBL properties on mean flow, with limited tunable parameters. • Able to get water to flow downslope with plausible levels of entrainment. • Weak dependence on vertical resolution (the BBL must be less than half the thickness of the bottom cell). • Concerns: • Pressure gradient errors drove upslope flow at the equator (Problem now fixed? G. Nurser pers. Comm.) • Does not work with partial cells. • May be pervasive in model code.

8. Song & Chao – Journal of Atmospheric and Oceanic Technology 2000 Embeds a Killworth & Edwards style variable thickness BBL within a fixed thickness (sigma-coordinate) slab appended to the bottom of a Z-coordinate OGCM. Particular attention is paid to the form of the pressure gradient terms, with claimed reduction of the pressure gradient error. • Strengths: • Has all of the strengths of Killworth and Edwards, plus it permits the use of partial cell topography. • Probably the best starting point framework for further refinements to the representation of BBLs in moderate to high resolution Z-coordinate models. • Concerns: • Pressure gradient errors may drive upslope flow at the equator. (Not tested.) • Certain important details vague in manuscript. • Not yet available in widely used model codes.

9. “Flow-specification” schemes Two such schemes already exist in MOM4. With partial cells, the impact on global climate models has been disappointingly limited. The strong dependence on vertical resolution can probably be overcome with modest additional complications. Relatively easy to implement as modular additions to existing models. NCAR has had favorable experience with Price & Barringer prescriptions for marginal sea inputs. The “injection”-driven circulation issues might be addressable by restructuring the order of calculations. “Flow-specification” schemes offer the best hope for near-term demonstration of the value of overflow representations for improving climate models. “Flow-specification” schemes may also have a future in low resolution climate models. Explicit BBL schemes No such schemes currently exist in the model codes in use for climate studies. GFDL experience with the Gnanadesikan scheme was disastrous for climate studies (5°C water at surface at equator!) with modest regional improvements. Improved explicit schemes need to be incorporated into the model codes. The pressure gradient errors must be addressed robustly for these to be of use. Explicit BBL schemes are the best hope for climate models that capture the physics underlying overflows and their dependence on the ambient circulation and mean state, but this will take more than 2 years to come to fruition. Developments in other coordinate systems will be most directly transferable with this approach. Modeling Center Perspective on Z-Coordinate BBL Approaches In BOTH cases, it is critical to test the schemes with the full suite of neutral physics used in climate models!

10. Bottom boundary layer mixing in isopycnal coordinate models • Resolution in isopycnal coordinate models automatically tracks the density front at the top of a gravity current. • A Richardson number dependent parameterization captures mixing observed in outflows. • Alternate work-based parameterizations can be readily adapted for use in isopycnal coordinate models

11. Constant Diffusivity Richardson Number Mixing

12. 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 (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 can almost certainly be improved upon.

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

14. Difficulties with Isopycnal Coordinate Modelsat Climate Resolutions Underresolved plumes tend to be broader and thinner than observed. • With a bottom drag applied as a body force, thin plumes entrain excessively. • Thin plumes may be embedded in viscous bottom boundary layers. • Local entrainment closures enable “double plumes” • The upper plume entrains vigorously and attains reasonable properties. • Dense water can slide downslope within the viscous bottom boundary layer. • If the bottom boundary layer is in a single-point wide channel, the velocity simply decreases, rather than rotating as it would in a well-resolved Ekman layer. Sill-exchange flows will be difficult to represent when the deformation radius and channel width are not resolved.

15. Modeling Center Plans for the CPT. The modeling centers envision an three-pronged approach to improving the representation of overflows in climate models within the scope of the present CPT. • A “plumbing” or “flow-specification” scheme will be developed in the short term for addition to existing Z-coordinate climate models. This prong will be used to show an impact of the CPT on current practice by the 2nd year review. • Isopycnal coordinate models will be coupled at GFDL within a year. Isopycnal models will be a focus for the near-term development of semi-resolved treatments of the overflows. • The modeling centers will implement and refine explicit BBL schemes in their ocean model codes. The developments from the second prong will be transferred to Z-coordinate models, starting within about 2 years.