Background Bulk exchange coefficient (Stanton Number) Flux parameterization

1. Heat and Salt Exchange at an Ice/Seawater InterfaceMiles McPhee, McPhee ResearchDavid Holland, CIMS NYU • Background • Bulk exchange coefficient (Stanton Number) • Flux parameterization • The “3-equation” solution • Evidence for double diffusion during melting • Is double diffusion important? • Freezing is different! • Special considerations for ice shelves

2. Before modern CTD and turbulent heat flux measurements were made under drifting ice, it was assumed that heat transfer was analogous to momentum transfer, which meant in essence that the mixed layer remained an ice bath at its freezing temperature, and that any heat entering the mixed layer, from below by mixing or from above by insolation, was transferred almost immediately to the ice cover. However, we knew from measurements (AIDJEX in the mid-‘70s) that mixed layer temperatures could remain above freezing (~0.1 - 0.2 kelvins) for months during the summer.

3. Elevation of mixed layer temperature above freezing, AIDJEX station Blue Fox, summer 1975 During the Marginal Ice Zone Experiment (MIZEX) in 1984, for the first time, we simultaneously measured mixed-layer properties, bottom ice ablation, and turbulent stress and heat flux near the ice undersurface.

4. A summary of the MIZEX measurements suggested a simple parameterization of basal heat flux in terms of the product of friction velocity at the interface and elevation of mixed layer temperature above freezing: where and cH is a bulk exchange coefficient, akin to a Stanton number (but based on friction velocity, not free-stream velocity). Its mean value for the MIZEX was about 0.006.

5. If kinematic stress is thought of in the same way as heat flux, i.e., proportional to a scale velocity times the change in velocity across the boundary: A typical value for cD under pack ice is thus about an order of magnitude greater than cH. Why? A more basic question might be: why does ice, which is nearly fresh, melt at all in polar water that is normally far below 0oC? In several experiments in both hemispheres since MIZEX, estimates of the turbulent Stanton number based on direct stress and turbulent heat flux measurements have yielded remarkably uniform values for cH; e.g., an average estimate from multiyear Arctic pack ice during the year-long SHEBA experiment in 1997-98 was cH = 0.0057+0.0004; while the corresponding value for smooth first-year ice in the Weddell Sea in 1994 was 0.0058+0.0013. The reason is that melting depends as much on salt dissolution as it does on heat transfer. At some small distance from the interface, molecular exchanges come into play and since the molecular diffusivities of heat and salt differ by a factor of about 200 in cold seawater, salt controls the melting process.

6. Note that when salinity flux is positive (ice melting), αH cannot be the same as cH, because S0 must be less than Sff. In fact, . ΔT is readily measured; δT is not, and this turns out to be an important distinction when melt rates are high We need to consider both the heat and salt balances at the interface. As before, express interface fluxes in terms of exchange coefficients times the product of a turbulent scale velocity and the difference between far-field properties (subscript ff) and interface properties (subscript 0): We further assume that water in a small control volume following the interface is at its freezing temperature, i.e., : , where m is the local slope of the “freezing line.”

9. If is small, as it often is in summer or in thick ice sheets, the two exchange coefficients enter the problem as a ratio, R = αH/αS, which indicates the strength of double diffusion, i.e., the tendency for heat to diffuse faster than salt. Based on laboratory studies and theory for laminar flow (the Blasius solution), R is thought to vary with the ratio of the Prandtl number to the Schmidt number raised to a fractional exponent, n. Given friction velocity, far-field T and S, and ice characteristics including conductive heat flux, the heat and salt balance equations may be combined for, e.g., a quadratic equation for S0:

10. Aside from the fact that ice melt rates indicate the importance of salt diffusion, is there evidence that double diffusion is important? • It is extremely difficult to measure heat flux, salinity flux, and Reynolds stress simultaneously near an ice/ocean interface. Nevertheless, a recent study (Sirevaag, Geophys. Res. Lett., 2009) presented direct estimates of αH (= 1.31×10-2) and αS (= 4.0×10-4) yielding R = 33. • A modeling study of migration of “false bottoms” that form during summer when fresh water collects under the ice, showed that the persistence and upward migration rates were highly dependent on double diffusion (Notz et al., J. Geophys. Res., 2003). Their results suggested a value of R closer to the high end of the range identified from laboratory studies (~70).

11. Most of the field measurements on which the empirical value for cH is based have been made in a limited range of forcing parameters, say But even if double diffusion is significant, does it warrant inclusion in larger scale ice/ocean modeling? Or stated another way, can we get by with constant cH? But often it is more extreme conditions that are of interest: for example, in a marginal ice zone with higher stress and larger ΔT, or under a tidally active ice shelf in contact with Warm Deep Water. If we stipulate that our chosen values for the αH and αS must satisfy the cH criterion within our measurement range, then the choice of R makes a large difference at more extreme forcing.

12. To illustrate this, consider the following conditions that we might typically measure under sea ice: u*0 = 0.015 m s-1, ΔT = 0.1 K, Hf = 34 W m-2 The bulk relation: yields this heat flux with cH = 0.0057 Now assume no double diffusion (R = 1) and determine the exchange coefficients that yield the same heat flux. Results: αH = αS = 0.0058, i.e., nearly the same as cH Same exercise, but with strong double diffusion (R = 70) Results: αH = 0.0144, αS = 2 ×10-4 , soall three approaches yield the same heat flux within the moderate parameter range Next, redo the exercise with ΔT = 2 K (slightly above 0oC) Bulk: cH = 0.0057, Hf = 680 W m-2(linear) R = 1 (no double diffusion), αH = 0.0058 ⇒ Hf = 681 W m-2(nearly linear) R = 70 (strong double diffusion), αH = 0.0144 ⇒ Hf = 988 W m-2 So strong double diffusion could result in nearly half again as much melt as the bulk relation in the presence of relatively warm water

13. Is double diffusion important when ice is freezing rather than melting? Recognition of the possible double-diffusive character of heat and salt exchange at the ice/ocean interface posed the possibility of significant supercooling and frazil ice production during freezing (Mellor et al., J. Phys Oceanog., 1986). In at least one ice/ocean modeling study, the effect was found to increase the equilibrium ice thickness, because frazil ice formed from double diffusion at the interface was distributed across all ice thickness categories, resulting in slower growth of the thinner categories, and enhanced overall heat loss (Holland et al., J. Geophys. Res., 1997). A subsequent series of experiments measuring fluxes under growing fast ice in tidally driven fjords showed that there was little or no tendency for supercooling or frazil production under moderate growth conditions, and that the freezing process is fundamentally different in character from melting (e.g., McPhee et al., J. Geophys. Res., 2008). We recommended that an algorithm that includes double diffusion and the quadratic solution during melting, say αH = 0.009 and R = 35, include a switch that sets R = 1 when the energy balance at the interface indicates ice growth.

14. Is what we know about sea ice applicable to ice shelf/ocean interaction? There is no a priori reason to suspect that shelf ice acts much differently from sea ice. However, there are several caveats: • Freezing temperature depends on pressure. As basal elevation changes, this may significantly affect ice/ocean interaction, increasing exchange when the interface slopes downward, and producing supercooling and marine ice when it slopes upward. An analogous “ice pump” mechanism thought to operate in sea ice differs by orders of magnitude. • Particularly near grounding lines, the underside may be crevassed and irregular on scales never seen in sea ice. We currently know little about representative hydraulic roughness for ice shelf undersurfaces, nor how to treat large scale fissures. • Large scale slopes in ice shelf basal topography may provide buoyancy forcing with no counterpart in floating sea ice.

15. Primary reference: McPhee, M. G., 2008: Air-Ice-Ocean Interaction: Turbulent Ocean Boundary Layer Exchange Processes, Springer, ISBN 978-0-387-78334-5, Chapter 6