The Steady Local Turbulence Closure Model
Abstract: A fundamental problem in boundarylayer physics is extrapolating limited measurements to a general description of the mean velocity and scalar properties, along with their Reynolds fluxes including values at the immediate boundary. For the atmospheric surface layer, extensive research has been devoted to methods relating relatively simple measurements to fluxes. Central to this approach is characterizing surface roughness for momentum and scalar variables. Typically, a tower is deployed with two or more levels of instrumentation and the surface fluxes are estimated either from the mean measurements across the tower using some form of the MoninObukhov dimensionless gradients (e.g., Businger et al. 1971; Andreas and Claffey 1995), or from a combination of mean gradients and fluxes, determined either by direct covariance or by spectral techniques (e.g., Edson et al. 1991).
In the IOBL, this is much less straightforward for a variety of reasons. First, in contrast to the upper seaice surface, variation in the underice morphology often occupies a significant fraction of the entire boundary layer. If the IOBL scales with about 1/30 of the atmospheric boundary layer, a pressure ridge with a 1m sail and 56m keel presents completely different aspects to the respective boundary layers. In general, for the IOBL parameterization problem, many of the surfacelayer assumptions (constant stress, stress and mean velocity collinear with no direction change, etc.) are clearly inappropriate.
As illustrated in Chapter 8, it is sometimes possible to solve a timedependent numerical PBL model with given initial conditions, letting it evolve in time as the forcing fields change. Given a suitable time series of observations at a particular location, to the extent that the model can reproduce the observed characteristics (say mixed layer temperature, salinity, depth), the model will provide a reasonably accurate description of the overall exchanges across the OBL. This depends on both having realistic initial conditions and a reasonably accurate time series of forcing fields (e.g., wind or ice velocity, conductive heat flux in the ice, etc.). In many cases, observations are scattered in both time and location (for example, stations taken from a ship or airplane during a regional survey), and one would like to produce a "snapshot" of the OBL structure, to estimate fluxes at the surface or near the base of the mixed layer.
M. McPhee, AirIceOcean Interaction, 173192. © Springer Science + Business Media B.V., 2008
Even when a relatively complete set of measurements exists, we are often faced with the sampling problem of extrapolating measurements made at a few location (which because of operational considerations, are often biased toward relatively smooth ice) to a general description for the entire surrounding ice field, which in turn might be appropriate to characterizing a grid cell in a numerical model (this is sometimes referred to as the "scaling up" problem). An example from ISPOL (McPhee 2008, in press) serves to clarify this problem. The floe with which we drifted north in the western Weddell Sea comprised a conglomerate of several different ice types including heavily ridged portions, relatively thin m) regions of first year ice, plus reasonably smooth regions of multiyear ice about 2 m thick. For most of the project the turbulence mast was located under ice of the last type, with the undersurface in the immediate vicinity quite smooth, but with pressure ridges and the floe edge within the first 100 m or so from the site. Toward the end of the drift phase of the project (on December 25) the ice floe split, forcing relocation of the turbulence mast, which for the last week of the project was located under thin ice near a small pressure ridge.
During the first deployment, we consistently observed a substantial increase in turbulent stress with depth across the 6 m span of the turbulence mast (see Fig. 9.10 of McPhee 2008), which we interpreted as the deeper sensors picking up turbulence generated by large undersurface features some distance away. This phenomenon has often been observed in other projects as well, typically where the mast was located under smooth ice, but there were roughness features within a distance given roughly by the ratio of mean velocity to scale velocity (u/u*) times the depth of the turbulence sensor measured from the interface (Morison and McPhee 2001). So, for example, a TIC 2 m below the boundary might sense roughness features within about 30 m, whereas turbulence measured 4 m lower might respond to undersurface protrusions up to 100 m away. This rule of thumb seemed to hold reasonably well for SHEBA as well as ISPOL (McPhee 2002).
For the short deployment at the end of the project, a mast with two clusters, 1 and 3 m below the ice undersurface, respectively, was initially placed so that the predominant tidal flow would approach from the north or south across relatively smooth ice, and parallel to a small pressure ridge situated to the west. Soon after deployment, however, the floe rotated so that if the current sensed by the mast came from the northeast, the keel was directly upstream from the turbulence mast, with large impact on flow in the upper few meters. The installation included an acoustic Doppler profiler that provided highresolution current profiles from about 10 to 30 m depth. Two examples from this second installation are shown in Fig. 9.1: one with flow approaching the mast across smooth ice, and the second with relative flow almost directly across the pressure ridge keel. In the former case, the current structure shows a reasonably well developed Ekman spiral, with friction velocity at 1 m of about 5mm s_1, and from direct application of the LOW to the dimensionless velocity, we infer a surface roughness of about 0.8 mm. When flow approaches from across the keel, the hodograph from ADP data in the range from 10 to 30 m again exhibits the expected Ekman turning, but now currents at the TIC depths are a small fraction of the deeper currents, indicating flow blockage. If u* estimated from the
Fig. 9.1 Currents measured relative to the drifting floe during the second deployment at ISPOL. The floe outline and orientation are shown, with mast location indicated by the square symbol and ADP current vectors at 10 m and 30 m depth. North is up. A more complete current hodograph with the TIC currents is shown in blowup view. Boxes list the friction velocity and dimensionless current. The heavy black curve represents the position of a small pressure ridge. a 3h average centered at time 365.25 (31 January 2004) with flow approaching from smooth ice, and b at time 363.5 with flow from across the ridge keel
Fig. 9.1 Currents measured relative to the drifting floe during the second deployment at ISPOL. The floe outline and orientation are shown, with mast location indicated by the square symbol and ADP current vectors at 10 m and 30 m depth. North is up. A more complete current hodograph with the TIC currents is shown in blowup view. Boxes list the friction velocity and dimensionless current. The heavy black curve represents the position of a small pressure ridge. a 3h average centered at time 365.25 (31 January 2004) with flow approaching from smooth ice, and b at time 363.5 with flow from across the ridge keel covariance statistics at 1m is taken at face value, Z0 is about 17 cm. With that much flow disturbance from the pressure ridge, many of the assumptions underlying the flux determinations would be suspect, and we would normally flag such data as unreliable. On the other hand, the example illustrates that form drag on pressure ridge keels will constitute a significant part of the total momentum transfer between the floe and ocean unless the undersurface is exceptionally smooth.
Evaluating the drag and enhanced mixing from even one pressure ridge keel is a formidable task requiring extensive computation (see, e.g., Skyllingstad et al. 2003), and extrapolating the results to an entire heterogeneous floe adds considerable difficulty. There is, however, a hint in the deeper current profiles in Fig. 9.1 that by considering what happens in the outer part of the IOBL, it may be feasible to infer surface properties representative of the entire floe, the main point being that because the floe moves as a rigid body, at depths greater than most of the undersurface protrusions, the turbulence must sense some integrated impact the varying surface conditions. In this chapter, we explore this concept with a modeling technique developed from the ISPOL measurements (McPhee 2008, in press).
9.1 Model Description
Unlike scalar conservation equations, the Ekman equation for momentum admits a steadystate solution. A "steady" version of the Local Turbulence Closure model (SLTC) was developed as a means of extrapolating limited measurements at particular times to deduce the structure of the entire boundary layer. The primary assumption and simplification for the SLTC model is that turbulence adjusts in effect instantaneously to surface conditions so that the local timedependent terms in the conservation equations are negligible relative to the vertical exchange terms (e.g., for momentum \ut \ ^ \tz — if u\) and that the vertical transport of TKE is not a major factor in most IOBL instantiations. While these assumptions are suspect when large inertial oscillation is present, or during rapid changes in surface flux conditions, they nevertheless often persist for reasonably long periods, especially when the ice cover is compact. In practice the model requires a reasonably good description of the temperature and salinity structure of the upper ocean, and some way of estimating friction velocity at the interface, perhaps from ice velocity or surface wind (if ice is drifting freely). As explained below, the model utilizes an iterative scheme that first estimates the IOBL eddy viscosity solely from surface flux conditions. Then by Reynolds analogy, it estimates scalar fluxes using eddy diffusivity based on the modeled eddy viscosity. In general, these fluxes will affect the turbulence scales and eddy viscosity, so the steady momentum is solved again with the new eddy viscosity, fluxes are recalculated, and so on. We demonstrated (McPhee 1999) that using this model to simulate the time evolution of temperature and salinity in the upper ocean produced results similar to a simulation using a secondmoment closure model (level 21/2 of Mellor and Yamada 1982). The latter required forward stepping of six conservation equations, while the SLTC time stepped only the T and S fields. In terms of computing cost, there was no great advantage, since the iterative scheme is computationally expensive, but the point was to show that the purely local (in space and time) turbulence description produced similar fluxes as the more sophisticated model which carried additional equations for momentum, TKE, and master length scale.
The model employs essentially the same physics as the timedependent model described in Chapter 7, except that rather than stepping forward in time from an initial state, forced by prescribed surface conditions, it considers a fixed upper ocean temperature and salinity state, with one set of interface flux conditions and iterates to a solution for momentum and scalar fluxes based on a physically reasonable distribution of eddy viscosity and scalar diffusivity.
9.2 The Eddy Viscosity/Diffusivity Iteration
Unlike the timedependent model, where for each time step the buoyancy flux and eddy diffusivities are determined from a previous time step, the "standalone" SLTC model begins from an initial guess at buoyancy flux, then iterates to a solution in which the modeled u* and observed T/S profiles determine the boundarylayer structure.
To illustrate the method, consider 3h average profiles of potential temperature and salinity from late in the SHEBA project (Fig. 9.2) and assume that u*o = 18mm s_1 is prescribed. This is used along with T and S in the upper ocean to calculate (w' b%. An initial guess for eddy viscosity (Fig. 9.3a) is made by determining a maximum value Kmax = u*oAmax where Amax is determined from u*o and w' b'} o according to the algorithm described in Section 7.6. An exponential falloff in stress is assumed except in the surface layer where it varies as K z u*o. This estimate assumes neutral stability throughout the water column, so that scalar diffusivity equals viscosity, which remains unrealistically large far past the mixed layer depth (indicated by the dashed line in Fig. 9.2b). As the arrow from a to b indicates, applying scalar diffusivity to the observed 0 and S profiles provides an initial estimate of buoyancy flux through the entire OBL, which is also unrealistically large below the mixed layer. By applying the mixing length algorithm with the first model estimate of profiles for (w' b'} (Fig. 9.3b) and u*o along with specified interface fluxes, a second Km estimate follows (Fig. 9.3c), from which new estimates are made (Fig. 9.3d), and so on, for a specified number of iterations. Results for eddy viscosity and buoyancy flux after the next iteration are shown in Fig. 9.3e and f, along with results after ten iterations (gray curves).
Details of the simulated eddy viscosity are shown in Fig. 9.4, along with estimates of eddy viscosity from two TICs, calculated from the products of local friction velocity and mixing length (inversely proportional to the wave number at
Potential Temperature Salinity
Potential Temperature Salinity

 Fig. 9.2 SHEBA profiler potential temperature and salinity from 3h average centered at 00:00UT on 20 September 1998, used to illustrate the SLTC model. The dashed line indicates the last grid point in the well mixed layer
the maximum in the w spectrum). In the upper part of the pycnocline (Fig. 9.4b), Km decreases exponentially with distance measured from the pycnocline depth, defined as the level at which squared buoyancy frequency first exceeds a minimum level, in this case 2.5 x 105 s2. A combination of u*p and (W b')p from the model solution at the pycnocline level determines X in the upper pycnocline. In the stable stratification of the pycnocline, the ratio of scalar diffusivity to eddy viscosity is a function of gradient Richardson number (Section 7.6) since turbulence is more effective at transferring momentum than scalar properties. This leads to a more rapid decrease with depth in Kh.
Modeled Reynolds stress, from the product of Km and the numerical velocity gradient, is shown as u* (the square root of kinematic stress magnitude) in Fig. 9.5a, along with the measurements. For this demonstration, u*0 was chosen so that the modeled stress matched measured at the lower instrument cluster (6 m below the ice) by successive adjustments to an initial guess assuming an exponential falloff from the interface to the 6m level. Modeled heat flux (—pCpKhQz), shown in Fig. 9.5b, indicates an upward flux of roughly 10 W m2 in the upper part of the well mixed layer, in agreement with measurements (pCp (W T') ) at the instrument cluster levels. Note that the only "modeled" part of the heat flux profile is the eddy diffusivity, Kh. The results are consistent with the interface flux dependent on the elevation of mixed layer temperature above freezing (square symbol). In the lower part of the
9.2 The Eddy Viscosity/Diffusivity Iteration Eddy Viscosity
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b 













m2s1 Eddy Viscosity
0 0.5 1 1.5 Wkg1 x 106 Buoyancy Flux: <w'b'>
m2s1 Eddy Viscosity
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0 0.5 1 1.5 Wkg1 x 106 Buoyancy Flux: <w'b'>
Fig. 9.3 a Initial eddy viscosity (Km) guess based on exponential ut profile and X calculated from surface fluxes. b Buoyancy flux profile from initial guess. c First iteration revised Km profile including buoyancy flux from b. d Second buoyancy flux estimate. e Second (black) and last (gray) eddy viscosity iterations. f Third (black) and last (gray) buoyancy flux estimates. Note the scale changes in buoyancy flux estimates
f 
















Wkg1
Fig. 9.3 a Initial eddy viscosity (Km) guess based on exponential ut profile and X calculated from surface fluxes. b Buoyancy flux profile from initial guess. c First iteration revised Km profile including buoyancy flux from b. d Second buoyancy flux estimate. e Second (black) and last (gray) eddy viscosity iterations. f Third (black) and last (gray) buoyancy flux estimates. Note the scale changes in buoyancy flux estimates
well mixed layer, the modeled heat flux is about twice as large indicating active mixing of heat from below. Presumably the flux divergence would heat the mixed layer as time progresses. This is, of course, an instantaneous snapshot, but indicates how the "steady" model may be used to estimate temporal evolution of upper ocean scalar properties (McPhee 1999). 
 Fig. 9.4 Eddy viscosity and thermal diffusivity after the iteration of Fig. 9.3, for the upper 40 m of the 80 m model domain a and detail in the pycnocline showing the reduction of scalar diffusivity relative to viscosity b
In the upper 10 m or so of the pycnocline (beginning at about 24 m) there is still relatively strong mixing of both momentum and heat, despite the rapid attenuation of eddy diffusivities because of upward buoyancy flux. Note that heat flux falls off in the pycnocline at about the same rate as momentum flux, even though the eddy thermal diffusivity is much smaller than eddy viscosity.
To recap, the demonstration shows that given measured profiles of T and S encompassing the well mixed layer and pycnocline, along with Reynolds stress measured at one level, a plausible distribution of momentum and scalar fluxes throughout the entire boundary layer may be constructed, including estimates of the interfacial fluxes. More information is required, however, to characterize the entire velocity structure (with respect to the undisturbed ocean velocity), namely, the undersurface hydraulic roughness, Z0. Generally, pack ice measurements are made from a platform that is moving relative to the underlying undisturbed ocean, and water velocity measured from the ice is not the absolute velocity in a fixedtoearth reference frame, but rather the vector difference between the absolute velocity at the measurement depth and the ice velocity. With modern satellite navigation, the latter may be measured quite accurately, and provided the orientation of the instruments is known (not always a trivial problem when dependent on compasses at high latitudes), it is a simple matter to determine the absolute velocity, say for example, u*=iti1/2
Fig. 9.5 a Friction velocity (square root of kinematic Reynolds stress) as modeled (solid) and measured at two levels. The surface value (square) symbol was chosen so that the modeled value at 6 m matched measured. b. Corresponding profiles and measurements of turbulent heat flux. The interface value (square) is calculated from uto, T and S in the upper ocean
6 m below the interface. But in a well developed, turbulent boundary layer, the 6m current comprises contributions from stressdriven shear in OBL, any inertial motion in the phaselocked ice/upper ocean system, plus the geostrophic current arising from slope in the sea surface. The last is the current that would exist without any shear between the ice and undisturbed ocean.
Since there is no provision in the SLTC model for inertial oscillations and in general the vector sum of geostrophic and inertial velocities at the measurement level is unknown, the surface roughness may be estimated from current measurement at a particular level as diagrammed in Fig. 9.6. The premise is that the topmost point in the mean quantity (zz) grid is within the surface layer so that surface stress and shear are aligned, in which case the velocity difference between the topmost grid point and the ice obeys the law of the wall:
Geometrically, Aw is determined by the intersection of an arc with length equal to the magnitude of the measured current (indicated by  Vm(rel) ), swung from the tip of the absolute model velocity vector (Vm(abs)) at the measurement depth, and a line extended in the direction of u*o from the velocity at the topmost grid point.
Measurement level
Measurement level
Fig. 9.6 Diagram showing how measurement of velocity at one level in the IOBL may be used to estimate the surface velocity and undersurface hydraulic roughness
Fig. 9.6 Diagram showing how measurement of velocity at one level in the IOBL may be used to estimate the surface velocity and undersurface hydraulic roughness 
 Fig. 9.7 The model hodograph from the September 20 example. Velocity measured relative to the ice at 6 m is used to scale the surface layer velocity, hence Vo. The ice velocity from satellite navigation, Vice, includes inertial and geostrophic shear effects not modeled. North is up
A vector extending from the model coordinate origin to this point then represents the ice velocity (¥0) relative to the underlying ocean in the model reference frame. The model is then oriented by aligning the modeled and measured relative current vectors at the measurement level. This entails multiplying horizontal velocity vectors by a complex factor:
The entire velocity solution for the example is diagrammed in plan view in Fig. 9.7. The model velocity relative to an observer on the ice matches the observed current r
Day 343.625
Day 343.625
Modeled Eddy Diffusivity
Modeled Eddy Diffusivity
30 psu
Fig. 9.8 SLTC model realization for 3h periods centered at 15:00 UT on 9 December 1997 during SHEBA. a model hodograph; b observed potential temperature and salinity profiles in model domain; c eddy viscosity/diffusivity. d friction velocity; and e turbulent heat flux
30 psu
d 

/ 






TICs 


Fig. 9.8 SLTC model realization for 3h periods centered at 15:00 UT on 9 December 1997 during SHEBA. a model hodograph; b observed potential temperature and salinity profiles in model domain; c eddy viscosity/diffusivity. d friction velocity; and e turbulent heat flux a
at 6 m, and establishes Vq, the vector ice velocity relative to the undisturbed ocean. This differs from the actual ice velocity obtained by satellite navigation (Vice) by "Vgeo" where quotes indicate that this is a combination of actual geostrophic flow plus any inertial or baroclinic motions, which are not considered in the SLTC model.1
One other example from early in the SHEBA project (Fig. 9.8) demonstrates a rare period during winter when there was downward turbulent heat flux in the water column, despite a lack of short wave radiation (the sun had set) and enough AT to imply a positive basal heat flux of about 1W m—2. There was a rather dramatic increase in stress from 8 to 12 m (Fig. 9.8d), probably from enhanced stirring by a pressure ridge keel about 110 m to the SW. Although not seen at the scale shown in Fig. 9.8b, there is a positive potential temperature gradient in the
1 "Vgeo" will also reflect any uncertainty in alignment of the turbulence mast, which often depends on compass headings and a model for magnetic declination.
well mixed layer that combined with the calculated eddy thermal diffusivity to produce the downward heat flux in the upper ocean, in agreement with the observations. A possible explanation for the positive temperature gradient is that all during the day of 9 December (day 343), as the ice drifted to the southwest, temperature of the well mixed layer decreased steadily by a total of about 12 mK (from — 1.493 to — 1.5o5 °C). From Fig. 9.8a, the absolute velocity hodograph shows that water in the upper part of the column was transported from NW (warmer) to SW (cooler) faster than in the lower column, resulting in a modest downward heat flux.
9.3 Applications
9.3.1 Ice Station Polarstern
The ISPOL project in the multiyear ice pack of the western Weddell Sea examined early summer airiceocean interaction from a wide range of physical, biological, and gasexchange perspectives (Hellmer et al. 2oo8, in press). A central issue was characterizing the turbulent exchange of scalar contaminants (including nutrients and biota) between the upper ocean and the sea ice during a time of maximum solar radiation. This presented a challenging problem in an ice pack with large variation in ice thickness, evidence of extensive ice deformation, and several embedded icebergs drifting with the sea ice within sight of the ship. Adding to the complexity was that, in contrast to summer pack ice in the central Arctic, the main driving force was not wind, but rather a combination of tidal and baroclinic currents just offshore of the eastern Antarctic Peninsula continental shelf.
As Fig. 9.1 suggests, applying measurements from a relatively smooth site to the entire floe may miss important parts of the momentum transfer as well as heat and salt exchange. Our ISPOL observations indicated that turbulent stress consistently increased with measurement depth, which meant that the effective surface roughness, zo, also increased with distance from the boundary (Fig. 9.9). There is rough agreement between zo estimates from the LOW and from a modification that considers "measured" mixing length (McPhee 2oo2), except for the shallow cluster at OTII, which was often affected strongly by flow blockage. The question posed is: which, if any, measurement level is representative of the entire floe? Currents measured by the acoustic Doppler profiler in the range from 1o to 3o m at both sites almost always showed counterclockwise (Ekman) deflection with increasing depth (in the drifting reference frame). We reasoned that these levels were below most of the obstructions on the ice underside, and that if currents were averaged over many directions and different current speeds, the resulting average current would reflect the integrated impact of varying undersurface morphology.
Of the 3h averages of ADP current profiles between 1o and 3om depth, there were 82 profiles (with acceptable signaltonoise ratios) where current speed at 3o m
Site OTI, 345.750 to 360.250 Site OTII, 362.625 to 367.500
246 Distance from ice, m
13 Distance from ice, m z0= 12.4 cm
246 Distance from ice, m
13 Distance from ice, m
Fig. 9.9 Median values of log(zo) for the main ISPOL turbulence mast site in December 2004. Squares are from a mixing length method (McPhee 2002); circles are application of the LOW. Error bars are 95% confidence limits (Adapted from McPhee 2008)
was greater than or equal to 0.06m s1. Each of these profiles was nondimensionalized by dividing the complex (vector) current at 2 m sampling intervals by the complex current at 30 m. The dimensionless current hodograph then produced a smooth spiral shape with about 15° of counterclockwise rotation as depth increased from 10 to 30 m (McPhee 2008, in press).
We reasoned that an estimate of overall surface roughness that was independent of the turbulence measurements and perhaps free from local topographic effects could be made as follows. First, specify a trial value for z00. Next, for each acceptable current profile, use the SLTC model to calculate the current profile. This was done by (i) specifying upper ocean T and S profiles, by interpolating in time from twice daily ship CTD stations; and (ii) forcing the model to match the measured current at 20 m, chosen to be generally in the well mixed layer, but deep enough to be away from the immediate impact of ridge keels. Since the object is to avoid using TIC data, for any particular model run, m*0 is first estimated from a Rossbysimilarity calculation, then adjusted iteratively until the model (relative to ice) velocity matches the ADP velocity at 20 m. We did this for three different values of zo, over the range of median values shown in Fig. 9.9. For each 3h model run (of 43 total), the modeled currents were nondimensionalized by the model 30 m current (in a reference frame attached to the ice), then averaged. Results (Fig. 9.10) indicate that turning between 10 and 30 m was best modeled with zo equal to approximately 4 cm, the value found for 4 m TIC at the main OTI site. This not only confirmed that the floe was relatively rough, but also provided a way of estimating the total floeaverage ocean/ice fluxes of momentum, heat, and salt during the entire ISPOL project.
Fig. 9.10 a. Average of 43, 3h average current hodographs divided by the current vector at 30 m (horizontal vector) for times when IV30I > 0.06m s_1. Vectors are drawn every 2m from 10 to 30m. The total Ekman angular shear between 10 and 30 m is P1030 = 14.6°. b, c Average model dimensionless hodographs for the same times with three different Z0 values (From McPhee 2008)
Fig. 9.10 a. Average of 43, 3h average current hodographs divided by the current vector at 30 m (horizontal vector) for times when IV30I > 0.06m s_1. Vectors are drawn every 2m from 10 to 30m. The total Ekman angular shear between 10 and 30 m is P1030 = 14.6°. b, c Average model dimensionless hodographs for the same times with three different Z0 values (From McPhee 2008)
9.3.2 Underice Hydraulic Roughness for SHEBA
A primary aim of the yearlong SHEBA project was to characterize multiyear ice in the western Arctic at scales useful for large scale modeling of airiceocean interaction, with particular attention to important terms in the surface energy budget. As discussed earlier, we often observed during the yearlong project that turbulence increased with increasing distance from the ice/ocean boundary. This suggested that deeper clusters were sensing upstream obstacles at increasingly distant fetch, including a prominent pressure ridge roughly 100 m away. We noted large variations in apparent roughness as drift direction and floe orientation varied over the yearlong deployment. By considering, stress, velocity, and apparent eddy viscosity at the TIC nearest the interface, we developed a technique that included X as inferred from spectral peaks as an independent parameter (McPhee 2002). This minimized the effect of upstream heterogeneity in the flow for determining the local hydraulic roughness of undeformed ice, and provided an estimate of around 6 mm for the undersurface hydraulic roughness. We emphasized that this value was not indicative of the "aggregate" floe roughness, which would include added drag from ridge keels and floe edges, or reduction from open water or smooth ice.
During the relocation of the SHEBA oceanography program after the floe breakup in March 1998, we drilled undeformed ice in numerous location, looking for slightly thicker hummocks from the previous summer melt period, obscured at the surface by the winter snow accumulation. These were thought to be the most likely locations for siting instruments and shelters that would survive the upcoming summer melt. We typically found about 20 cm difference between ice thickness in hummocks versus "fossil" melt ponds. Laboratory studies show that hydraulic roughness is typically about 1/30 the "grain size" of the elements contributing roughness. For the portions of the SHEBA floe away from pressure ridges and leads, Z0 = 6 mm would imply a "grain size" of about 20 cm, hence is not inconsistent with our limited observations.
The question remains: what is the "aggregate" roughness of a typical multiyear Arctic ice floe in the region traversed by SHEBA? The previous analysis (McPhee 2002) utilized the cluster nearest the interface (nominally 4 m until summer, when it was raised to 2 m), and used an estimate of the mixing length there to adjust shear between the interface and measurement level, which tended to decrease Z0 from its LOW value at SHEBA site 1 (November to midMarch), and increase it slightly at site 2 (midMarch through September) relative to the LOW estimate. Measurements at the former site were obviously affected by a pressure ridge keel that was often "upstream" as the station drifted west and later north, while site 2 was farther from any apparent features. When averaged over the entire site 1 deployment, there was a monotonic increase in average u* with depth from clusters 1 to 3 (nominally 412 m from the ice).
To address the SHEBA "scaling up" problem, we speculated that a technique analogous to that developed to characterize the ISPOL floe could be adapted for the SHEBA data. The approach settled on was somewhat different. Although there was an acoustic Doppler profiler at SHEBA, its data return rate over the course of the deployment was disappointing, and there often appeared to be spurious returns in the upper portion of the current profiles that contaminated measurements within the well mixed layer. We chose instead to use Reynolds stress and current measurements from TIC 2 on the turbulence mast, along with T/S profiles from the SHEBA automated profiler to solve the SLTC model for each 3h average in the period from 15 November 1997 to 1 June 1998 when ice drift speed (without the inertial component) exceeded 0.1ms_1. Cluster 2 was chosen because it was at a depth (nominally 8 m from the ice) thought to minimize the impact of upstream heterogeneity and because it had the most samples (clusters 3 and 4 were sometimes below the well mixed layer, and were not redeployed after the March 1998 breakup).
A time series of log(z0) for each of 249, 3h model realizations meeting the minimum velocity requirements (also excluding about 24 samples where the derived Z0 was smaller than 6 mm) is shown in Fig. 9.11, along with averages in tenday bins. The mean value with standard deviation error bars is log(z0) = —3.0± 1.0. The mean value is thus about 4.9 cm with a range implied by the standard deviation oflog(zo), 1.6 < Zo < 14.6cm.
The dimensionless surface velocity r = ¥0/^*0, shown in Fig. 9.12 for each model realization, is the complex inverse of a "geostrophic drag coefficient" that includes turning angle as well as magnitude. In terms of a more conventional quadratic drag coefficient, cw (where T0 = u*02 = cwv02), the mean magnitude of r implies cw = 0.0056. This is nearly equal to the value of 0.0055 derived in a different manner from the freedrift force balance for the AIDJEX stations in the central Beaufort Gyre in 1975 (McPhee 1980). The latter depended on a relatively high value for the 10m wind drag coefficient based on analysis of balloon soundings during AIDJEX (Leavitt 1980; Carsey 1980). The mean turning angle inferred from the AIDJEX freedrift force balance was slightly less: 23°.
9 The Steady Local Turbulence Closure Model Model Derived log(z0)
320 340 360
380 400 420 440 460 Year Day where 1 Jan 1997 = 1
480 500 520
Fig. 9.11 Time series of the natural logarithm of underice surface roughness derived from SLTC model runs based on measured Reynolds stress and relative velocity at TIC 2 on the SHEBA turbulence mast, nominally 8 m below the ice/ocean interface (see also Colorplate on p. 212)
320 340 360
380 400 420 440 460 Year Day where 1 Jan 1997 = 1
480 500 520
Fig. 9.11 Time series of the natural logarithm of underice surface roughness derived from SLTC model runs based on measured Reynolds stress and relative velocity at TIC 2 on the SHEBA turbulence mast, nominally 8 m below the ice/ocean interface (see also Colorplate on p. 212)
Dimensionless Surface Velocity
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320 340 360 380 400 420 440 460 480 500 520
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* 
+ 
SL + 
It 
I; 
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* 
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+ 
* *+ 
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t 
320 340 360 380 400 420 440 460 480 500 520
Day of 1997
Fig. 9.12 a Magnitude of dimensionless surface velocity (with respect to undisturbed flow) for the model solutions. b Magnitude of the angle between Fq and uto a
In contrast to the AIDJEX analysis, the magnitudes of r and P do not decrease with increasing Vq as would be expected if the boundary layer strictly followed Rossby similarity scaling. Indeed, the trend is opposite, with r increasing in both magnitude and deflection angle. The most plausible explanation is that during almost all of SHEBA the well mixed layer was significantly shallower then during corresponding time at AIDJEX. In similarity terms, the nondimensional pycnocline depth was thus particularly small for higher speeds during SHEBA, so that momentum flux was more confined vertically which tends to increase both the dimensionless surface velocity and the amount of OBL turning.

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