1 Ice-Ocean Stress I. Quadratic drag laws: background, physics a. 2-d vectors as complex numbers b. Free-drift force balance c. Drag coefficients II. IOBL similarity a. Dimensionless variables b. Ekman stress equation c. Ekman “surface” velocity d. Combine Ekman and surface layers e. Rossby similarity-- why? III. Impact of stratification a. Rapid melting-- generalized Rossby similarity b. Ice edge bands c. Shallow mixed layers IV. Recommendations a. Model considerations b. Undersurface hydraulic roughness McPhee, M. G., 2011: Advances in understanding ice-ocean stress during and since AIDJEX, Cold Reg. Sci. Technol., doi: /j.coldregions

2 2-D vectors as complex numbers x (u) y (v) 

3 Force Balance 1: typical of AIDJEX z 0 = 0.07 m Ice Thickness: 3 m c 10 = |V ice |/U wind =1.9% c w = e i22° Force Balance 2: typical of ANZFLUX z 0 = m Ice Thickness: 0.5 m c 10 = |V ice |/U wind =3.3% c w = e i13°

4 AIDJEX Pilot, h average on 12 Apr (McPhee and Smith, JPO, h average on 1 Apr

5 Steady, horizontally homogeneous boundary layer equation: Constitutive law relating stress in the fluid to shear (Ekman postulated that eddy viscosity was independent of depth): Boundary conditions: stress at surface specified, velocity vanishes at depth: 2 nd order ordinary differential equation: The Ekman Layer

6 Ekman’s solution for the steady, unstratified boundary layer forced by stress at the surface of a deep ocean. The net volume transport is perpendicular to the surface stress

7

8 Near the boundary in a neutrally stratified shear flow, the velocity profile is logarithmic, and is described empirically by: The Surface Layer

9 The Surface Layer (continued)

10 Seek by suitable choices for scales, to reduce a whole class of fluid dynamical regimes (in this case, neutrally stratified, rotating planetary boundary layers) to a common set of equations, with one solution. Similarity

11 Similarity (continued)

12 Dimensionless Ekman velocity Similarity (continued)

13 Surface Layer Ekman Layer Similarity (continued)

14 So now we can put together an “inverse drag law:” Similarity (continued)

15 Γ cwcw β Similarity (continued)

16 Rapid Melting Shallow Mixed Layers Internal Waves Impact of Stratification

17 Stratification (continued) Extend the similarity theory to include buoyancy flux from rapid melting at the interface

18 By analog with the neutral Rossby similarity development, formulate an expression for the dimensionless surface velocity: McPhee, M. G, 1981: An analytic similarity theory for the planetary boundary layer stabilized by surface buoyancy, Boundary-Layer Meterol., 21, McPhee, M. G., 2008: Air-Ice-Ocean Interaction: Turbulent Ocean Boundary Layer Exchange Processes, Springer, ISBN Stratification (continued)

19 The impact of rapid melting is to reduce the turbulent length scale, and increase the velocity scale. It increases both A and B, which reduces the effective drag and increases the turning angle. The rapidly melting ice (water about 2 o C) drifts about 6.5 km farther in a day. Stratification (continued)

20

21

22 Stratification: Shallow Pycnoclines In the western Arctic, there has been a remarkable increase in freshwater content over the past decade. This tends to stratify the water column closer to the surface, so the depth of the mixed layer adds a 4th length scale to the IOBL turbulence.

23 To address this part of the problem requires treatment of fluxes in the upper part of the pycnocline, which are not amenable to the simple similarity approach. However, a first-order turbulent closure model I call steady local turbulence closure (SLTC) uses the same similarity principles.

24 When stress is mild, the impact of a shallow (20 m) mixed layer on surface velocity and drag is minor compared with one 2½ times as deep. At higher stress, the difference is more apparent, manifested mainly as increased turning angle

25 The SLTC model can also be used to examine a perennial problem with measurements made from drifting ice: how to scale results up to represent an entire floe or grid area in a numerical model. During the ISPOL project in the western Weddell ( ) we measured acoustic Doppler profiler currents in the upper ocean. I took every 3-h average, divided by the complex velocity at 30 m and then averaged the results for each depth for most of the project. Large-scale Hydraulic Roughness

26 Then model each 3-h segment using the measured T/S structure from the ship and matching the 20 m current. Each modeled profile is again nondimensionalized by the 30 m current, and averaged

27 Locationz0z0 Type ISPOL (western Weddell) 40 mmMultiyear pack iceMcPhee, Deep-Sea Res., 2008, doi; /j.dsr SHEBA (western Weddell) 49 mmMultiyear pack iceMcPhee, Air-Ice-Ocean Interaction, 2008 NPEO (North Pole)90 mmMultiyear pack ice, highly deformed Shaw et al., JGR, 2008, doi: /2007JC MaudNESS (eastern Weddell) 4 mmThin, first year iceSirevaag et al., JGR, 2010, doi: /2008JC SLTC Roughness Estimates

28 Other Roughness Estimates McPhee, M.G., 1990: Small scale processes, in: Polar Oceanography, ed: W. Smith, Academic Press,

29 Recommendations for Drag Parameterization Recognize that drag coefficient and turning angle ( β ) depend on v 0. Especially for thick ice, this can change the effective drag by as much as a factor of 2 over a reasonably speed range. For deep mixed layers and slow melting use Rossby similarity. If water is warm, allow for stratification effects, either by incorporating them into a good IOBL model, or by applying the modified Rossby similarity. This may increase ice divergence, both in MIZ’s and where low concentration insolation has been intense.

30 Recommendations for Drag Parameterization If the pycnocline is shallow, recognize that it may appreciably change β in the opposite sense from Ro * similarity; e.g., expect larger β at higher speeds. There is less impact on drag magnitude. 47 m pycnocline 20 m pycnocline

31 Recommendations for Drag Parameterization Regarding z 0. ‣ For “normal” multiyear pack ice: mm is a good guess. ‣ In highly deformed ice and MIZ’s: mm. ‣ For drifting first year ice, with minor deformation: 1-4 mm. ‣ For undeformed fast ice: hydraulically smooth, unless there is platelet growth. ‣ Mixed first-year/multiyear: Use weighted average of the logarithms of z0 for the first and multiyear fractions. Internal Waves: These were observed to affect momentum flux during the 1984 MIZEX project. Details and a suggested parameterization are given by: McPhee, M.G., and L.H. Kantha, 1989: Generation of internal waves by sea ice, J. Geophys. Res., 94, , 1989.