mcphee 1992

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 97, NO. C4, PAGES 5365-5379, APRIL 15, 1992

Turbulent Heat Flux in the Upper Ocean Under Sea Ice

MILES G. MCPHEE

McPhee Research Company, Naches, Washington

Turbulence data from three Arctic drift station experiments demonstrate features of turbulent heat transfer in the oceanic boundary layer. Time series analysis of several w' T' records shows that heat and momentum flux occur at nearly the same scales, typically by turbulent eddies of the order of 10-20 m in horizontal extent and a few meters in vertical extent. Probability distribution functions of w'T' have large skewness and kurtosis, where the latter confirms that most of the flux occurs in intermittent "events" with positive and negative excursions an order of magnitude larger than the mean value. An estimate of the eddy heat diffusivity in the outer (Ekman) part of the boundary layer, based on measured heat flux and temperature gradient during a diurnal tidal cycle over the Yermak Plateau slope north of Fram Strait, agrees reasonably well with the eddy viscosity, with values as high as 0.15 m 2 s -1 . An analysis of measurements made near the ice-ocean interface at the three stations shows that heat flux increases with both temperature elevation above freezing and with friction velocity at the interface. It also reveals a surprising uniformity in parameters describing the heat and mass transfer: e.g., the thickness of the "transition sublayer" (from a modified version of the Yaglom-Kader theory) is about 10 cm at all three sites, despite nearly a fivefold difference in the under-ice roughness z0, which ranges from approximately 2 to 9 cm. A much simplified model for heat and mass transfer at the ice-ocean interface, suggested by the relative uniformity of the heat transfer coefficients at the three sites, is outlined.

1. INTRODUCTION

A series of experiments performed over the past few years aimed at measuring turbulent fluxes in the boundary layer under drifting pack ice has provided a relatively comprehen- sive view of the processes by which heat is transferred in the oceanic boundary layer. This is significant not only for predicting heat and mass transfer at the ice-ocean interface, but also for understanding how turbulence distributes scalar properties in a rotational boundary layer. Few question, for example, Reynolds' analogy in the well-mixed layer, i.e., that the turbulent exchange coefficient for scalars is essen- tially the same as for momentum; yet one is hard pressed to cite actual experimental evidence that it holds. In the open ocean mixed layer, scalar and momentum fluxes are nearly always inferred from measurements of other terms in the conservation equations; few, if any, actual direct flux measurements exist because of the difficulty of measuring vertical turbulent velocities in a surface wave field, (Weller et al. [1985] reported measurement from research platform FLIP of vertical velocities associated with Langmuir circulations, but they did not estimate turbulent Reynolds stress.)

Oceanic turbulence measurements are usually made at small scales, near the dissipation range of the turbulent spectrum, using microshear, temperature, and conductivity probes mounted on profiling sondes [e.g., Gregg, 1987, 1989; Padman and Dillon, 1991] or on submarines [e.g., Osborn and Lueck, 1985]. In some ways, this is a "mature technol- ogy"; yet many questions remain about the various assump- tions (e.g., homogeneity, isotropy, neglect of various terms in the turbulent kinetic energy and temperature variance budgets) and techniques involved in going from measurements near the Kolmogorov microscale to the actual fluxes. Yamasaki et al. [1990] report nearly a factor of 2 difference in dissipation rates measured at the same place and time via

Copyright 1992 by the American Geophysical Union.

Paper number 92JC00239. 0148-0227/92/92JC-00239505.00

a horizontally sampling submarine and a vertically profiling instrument; they attribute the difference to undersampling by the vertical probe. Recently, Moum [1990] has called into question the sampling problem from a different perspective. He used a sensitive Pitot tube arrangement to measure the deviatory vertical velocity w' along with microshear from a profiling sonde and found that in many turbulent patches in the thermocline, w'T' (where T' is the deviatory temperature) was countergradient. There was a large disparity in the heat flux calculated as the average of w' T' actually measured in 40 patches versus heat flux calculated from an eddy coefficient based on the dissipation rate. The latter is necessarily down the mean gradient. Moum questions whether high dissipation rates in "restratifying" parts of the eddies (where w' T' is countergradient) might not unduly influence scalar diffusion coefficient estimates. In the mixed layer, almost by definition the mean scalar gradients are extremely small, which exacerbates sampling of quantities involving the local buoyancy frequency. All of these factors point to the need for direct flux measurements.

In contrast to the open ocean, it is relatively easy to measure turbulent velocities below drifting ice floes. There is by now a fairly large body of turbulence data, beginning with the pioneering work of Smith [1974] on the under-ice rotational boundary layer during the Arctic Ice Dynamics Joint Experiment (AIDJEX). The accumulated data include several examples of Ekman spirals in turbulent stress as well as mean velocity [McPhee, 1990]. During the 1984 Marginal Ice Zone Experiment (MIZEX 84), fast response thermometers and conductivity meters were added to the Smith current meter clusters. McPhee et al. [1987] described the basic experimental setup and showed the correspondence of measurements of turbulent heat flux in the under-ice boundary layer with estimates of heat flux based on ablation of the ice underside. We also demonstrated the importance of the molecular effects in thin sublayers near the interface, in keeping with a theory for turbulent exchange over hydraulically rough surfaces developed by Yaglom and Kader

5365

5366 MCPHEE' TURBULENT HEAT FLUX IN THE UPPER OCEAN UNDER SEA ICE

TABLE 1. Scalar Mean and Turbulent Flow Properties for Three 1-Hour Segments of Data at Two Levels Below

the Ice, From MIZEX 84

Day

Speed,

Depth, cm T- TT, u ,, cm q, p c p ( w ' T2'), m s -• K S -l cm s -l W m-

189 4 15.91 0.18 1.05 2.63 18.7 7 17.54 0.19 0.90 2.61 5.8

180 4 9.51 0.24 1.26 2.84 87.3 7 12.60 0.26 1.20 2.63 112.3

191 4 9.04 1.21 0.92 2.83 374.9 7 11.91 1.25 1.04 2.84 479.5

[1974]. Since MIZEX, two additional experiments that were part of the yearlong Coordinated Eastern Arctic Experiment (CEAREX) have extended the turbulent heat flux data base.

The present work presents a partial synthesis of heat flux data from the three Arctic experiments: MIZEX 84, a summer drift of about 180 km centered near 80ø20'N, IøE; CEAREX 88, an early winter drift of about 200 km centered near 81ø50'N, 36øE; and CEAREX 89, a late winter drift of about 150 km centered near 82ø20'N, 8øE. In the next section, fundamental aspects of turbulent heat flux near the ice-ocean interface are demonstrated with some 1-hour data

segments covering a wide range of heat flux magnitudes, partly to show that the turbulence measuring system can adequately measure vertical heat flux, but mostly to look at the time and space scales and the form and distribution of w' T'. The intent of section 2 is to analyze a few "snapshots" of the turbulent flow in order to lay groundwork for the more extensive statistical treatments that follow. Section 3 con-

siders a particular diurnal tidal cycle of turbulent fluxes deeper in the mixed layer over the northern flank of Yermak Plateau during the CEAREX 89 drift. This 1-day segment illustrates much about turbulence in the outer or Ekman part of the boundary layer, including independent measurements of eddy viscosity and eddy heat diffusivity (which, inciden- tally, conform with Reynolds' analogy). Section 4 considers averages of heat flux as a function of surface stress and elevation of mixed layer temperature above freezing, to estimate the parameters necessary for heat and mass transfer models appropriate to the ice-water interface. Results are summarized in section 5.

2. EXAMPLES OF TURBULENT HEAT FLUX

MEASUREMENTS

Since direct measurements of heat flux in the ocean are

relatively scarce, the purpose here is to examine several segments of data from the near-surface boundary layer under drifting ice to demonstrate some characteristics of the turbulent heat transfer process. The turbulence clusters com- prise three small, partially ducted rotors arranged in a horizontal plane near fast response temperature and conductivity sensors. Two of the rotors are mounted along axes facing 45 ø up and down from the horizontal, both orthogonal to the third axis in the other horizontal direction. Frequency output from each instrument in the cluster is sampled 6 times per second, then typically averaged for l-s periods and recorded digitally. A more complete description of the turbulence cluster (TC) system, including velocity calibration details, is given by McPhee [1989].

During the summer 1984 M1ZEX drift in the marginal ice zone of the Greenland Sea, we encountered mixed-layer temperature conditions ranging from near freezing to periods of relatively rapid bottom melting in water temperatures more than a degree above freezing. For MIZEX 84, we deployed an inverted mast with five clusters suspended at 1, 2, 4, 7, and 15 m below the ice underside near the center of a large floe, with a sixth TC deployed at 2 m, about 100 m away. A description of upper ocean measurements, including a comparison of turbulent heat flux measurements with underside ablation, is given by McPhee et al. [1987]. For the present purposes, data from the clusters 4 and 7 m below the ice are considered, because these depths were consistently within the mixed layer, but beyond the immediate surface layer where local under-ice topography influenced the turbulence. One-hour segments of data from each of these clusters on three different days exemplify a wide range of upward heat flux. Mean quantities are summarized in Table 1. The turbulent (scalar) velocity scales are defined by u, = ({//'W'} 2 + {Z•'W'}2) 1/4 andq -- ((u '2) + (v '2) + {w'2}) 1/2, with the prime denoting a deviatory quantity.

Time series of temperature, salinity, and velocity are shown in Figure 1 for cluster 4 (7 m) during a 1-hour period on day 180 of 1984 (run 180.01). The data are plotted as deviations from the mean, which is listed at the right for each variable. The original velocity components have been rotated into a reference frame chosen so that the mean u

component is eastward, mean v is northward, and mean vertical velocity vanishes. This is a fairly typical record of turbulent flow in the ice-ocean boundary layer. Autospectra of the w' and T' time series of Figure 1, normalized by the total respective variances, are shown in Figure 2, along with squared coherency. The spectra were calculated by smoothing the periodograms (i.e., squared magnitude of the Fourier transforms) with four passes of a modified DanJell (boxcar) smoother of half width 4 s-I [Bloomfield, 1976]. Squared coherency, which is calculated by dividing the squared magnitude of the smoothed cross periodogram by the product of the smoothed autospectral estimates, indicates via the

720 1440 2160 2880 3600 sec

I ' ' I ' ' I ' ' I ' ' I

0.02 • T(de• 0 <-1.58>

0.02

0 <33.45>

5.0 u (cln/s)

0 <-2.58>

5.0 • v (cm/s) 0 <12.34>

5.0

• w (cm/s) 0 <0.0>

Fig. 1. One-hour time series of temperature T, salinity S, and u (east), v (north), and w (vertical) velocity for cluster 4 (7 m below the ice) on day 180 of 1984. Series are plotted as deviations from the mean value, listed to the right.

MCPHEE' TURBULENT HEAT FLUX IN THE UPPER OCEAN UNDER SEA ICE 5367

real part of the cross spectrum at what frequencies the two time series are cross correlated.

The similarity of the w' and T' autospectra, particularly the -5/3 slope region encompassing the inertial subrange for high Reynolds number turbulence, shows that the turbulent variation in heat content is similar to that for momentum. If

we accept that frequency is proportional to wavenumber (the frozen field hypothesis) and that the scale of the largest (energy containing) turbulent eddies is indicated by wavenumbers just below the start of the -5/3 energy cascade, then the similarity of the spectra shows that temperature variance "dissipation" occurs at nearly the same scales as turbulent kinetic energy (TKE) dissipation. By analogy, heat flux and momentum flux (stress) should occur at similar scales. The same is indicated by the squared coherency, showing that significant correlation occurs predominantly at lower frequencies (wavenumbers), and a broad peak around f = 0.01 s -1 suggests that the main turbulent flux is accomplished by eddies with time scales of the order of minutes and horizontal length scales of the order of tens of meters.

Spectra of the actual w'T' time series at both 4 and 7 m are shown in Figure 3a. The -5/3 slope persists, indicating that the" energy containing" eddies are also the ones responsible for most of the heat flux. There is little coherency at any frequency between the two levels, suggesting that the main "heat flux events" are limited, at least in this case, to less than 3 m in vertical extent. A similar plot of the downstream (northward) velocity deviation spectra at 4 and 7 m is shown in Figure 3b. There is significant coherency in velocity for features with time scales greater than about 2.6 min; however, Figure 2 shows that the main w', T' coherency occurs at higher frequencies; thus the lack of coherency between w' T' measured at the two levels is not surprising.

We next consider instantaneous fluxes of momentum and

heat in the time domain directly. Time series of u'w', v'w', and w'T' taken from the one-second averaged data are shown in Figure 4, with the kinematic Reynolds stress and heat flux averages listed at the right. (At the current speeds measured, the rotors mechanically filter motions with periods shorter than 1 s.) Here the velocity components of cluster 4 have been rotated into a frame of reference where

the u ("downstream") component is in the direction of the mean flow so that the v ("cross stream") component has zero mean. As expected in a rotational boundary layer, a vector formed from the horizontal Reynolds stress components is to the right of the mean velocity. Note that the instantaneous w'T' series is similar in appearance to the instantaneous Reynolds stress, with many negative and positive excursions, reminiscent of the reversals of instantaneous w' T' values measured in turbulent patches in vertical profiles, as reported by Mourn [1990].

A question closely related to the scale of the "energy containing" eddies is what effect averaging (or low-pass filtering) has on determining heat flux. To investigate this, (w' T') was calculated for each of the six data segments after first low-pass filtering the w' and T' time series with several different values of the cutoff frequency, ranging from 1/5 to 1/30 s -• . The resulting fluxes, expressed as a percentage of the unfiltered (1 s) flux, are shown in Figure 5, along with a least-squares-fitted, second-order polynomial R = 1.006 - 3.9 x 10-3t•, - 4.3 x 10-5 t•. This suggests that rapid sampling is not necessary for capturing the bulk of the w', T'

1.0

-1.0

-2.0

-3.0

-4.0 -4.0

1.0

0.5

0 -4.0

I95%

(w'w') = 2.31

(T'T') = 3.79x10 -5

-3.0 -2.0 -1.0 0

Squared Coherency

-3.0 -2.0 -1.0 0

log (f)

Fig. 2. Log-log plot of power spectral density divided by total variance, for l-hour time series of w' and T' from Figure 1. Frequency f is in cycles per second. The temperature spectrum is displaced I decade downward for clarity. Total variances in cm 2 s-2 and K 2 are listed on the plot. The bottom panel shows squared coherency for the two time series. Values below the dashed line are not significantly different from zero at the 95% confidence level [Bloomfield, 1976].

covariance: for instance, even if all fluctuations with periods less than 20 s are removed from a flow of 10-15 cm s -• more than 90% of the heat flux remains.

Time series of w'T', formed after filtering the original series with a cutoff frequency of 0.1 s- 1 are shown in Figure 6. Mean values, plotted as dashed horizons with magnitude listed at the right, are typically an order of magnitude smaller than the size of the positive and negative excursions. The scales for each day are roughly 10 times the mean value and differ by a factor of 33. Despite the range of heat flux encountered, the time series appear similar, confirming that the scale of the turbulent eddies responsible for heat trans- port depends mainly on shear forcing, not thermal forcing. The graphs also illustrate the practical aspect of why it takes quite a while for turbulence statistics to "settle down'" even over an hour, one major excursion of several minutes' duration can have a large impact on the mean, which is relatively small in magnitude.

These examples are fairly typical of turbulence measurements under sea ice when the mixed layer is above freezing. The heat flux values, ranging up to about 500 W m -2, span most commonly encountered conditions, although at the extreme ice edge, reported melt rates as high as half a meter per day imply much higher localized heat flux. In that case, buoyancy from melting ice may substantially alter the boundary layer turbulence [McPhee, 1983].

Inspection of Figure 6 reveals that the time series of instantaneous heat flux have common characteristics, including what appears to be asymmetry in the positive and negative excursions, plus a tendency to remain near zero between separate "events." Simple statistics of the six time

5368 MCPHEE: TURBULENT HEAT FLUX IN THE UPPER OCEAN UNDER SEA ICE

1.0

-4.0 -4.0

95%

02 (4m) = 3.66x10 -5 w'T'

02 (7m) = 8.52x10 -5 w'T'

-3.0 -2.0 -1.0 0

Squared Cohercncy

0.5

0 • • • • I ' -4.0 -3.0 -2.0 -1.0 0

log (0

1.0

-1.0

-4.0 -4.0

1.0 .

0.5

o -4.0

'3

B

95%

(v'v')a,n = 4.66

(v'v')?,. 4.44 , , ,, I, , •, I , , , , I , ,•

-3.o -2.o _1.o -- o

Squa.nxl Coherency

-3.0 -2.0 -1.0 0

log (0

Fig. 3. (a) Normalized power spectra for the one-hour time series of w'T' for clusters 3 and 4 at 4 and 7 m, respectively, in the same representation as Figure 2. In this case, there is little cross correlation between the two series at any frequency, although they are separated by only 3 m in the vertical. (b) Normalized power spectra for the one-hour time series of northward velocity deviations (i.e., in the main flow direction) as in Figure 3a. There is little coherency at periods shorter than about 2.5 min.

series are listed in Table 2, where the third and fourth moments, skewness and kurtosis, are nondimensionalized by the sample standard deviation •r, i.e., the square root of the second moment. Skewness indicates departure of the probability distribution function (pdf) from symmetry, with positive values implying a gentler slope in the positive direction. Kurtosis is a measure of the flatness of the pdf relative to a Gaussian distribution: an abrupt peak has large positive kurtosis. While there is much variability in the higher-order moments, all are much larger than what might be expected from natural variation of a Gaussian pdf. The mean kinematic heat flux, nondimensionalized by the standard deviation, is also shown in Table 2; although the sample size is small, there may be a significant difference here

10.0

0.0

o 10.0

0.0

0.05

0.0

0 600 1200 1800 2400 3000 3600 sec

I''''''''' ''"''"''1''''"'$'1" ''''' ''1 '•''''''' I"'''"''1

"downstream"

i U"W' <u'w'> = 1.27

(w'T') = 2.75x10 -3

Fig. 4. Time series of kinematic Reynolds stress and heat flux for cluster 4, 7 m below the ice, on day 180 of 1984. The reference frame is rotated so that u is in the direction of the mean streamline.

Multiply w' T' by 4.1 x 10 4 for heat flux in watts per square meter.

between the two levels, perhaps related to the size of the largest eddies.

The probability distribution functions for each time series were approximated by smoothing 50-bin histograms of w'T'/o' with a cubic spline, then averaging the results for each cluster depth (Figures 7a and 7b). Nondimensionaliz- ing by the sample standard deviation shows that the shapes of the estimated pdf's are all similar. In fact, a remarkably simple expression made from two exponential segments emanating in positive and negative directions from the

105

100

Smoothing Period, s

Fig. 5. Heat flux as a function of low-pass filtering period applied to the temperature and vertical velocity time series, expressed as a percentage of the heat flux with no filtering (i.e., from l-s averages). Symbols show the percentages for six l-hour averages, at 5-s intervals in the smoothing period, tp.

MCPHEE: TURBULENT HEAT FLUX IN THE UPPER OCEAN UNDER SEA ICE 5369

......... ......... ......... ......... ] gO min,,,,,,,, • I • •

o.oo3 (^)

Day 189 0 Ave= 4.27e-04

-0.oo3

0.03

0 Day 150 Ave= 2.05e-03

-0.03

0.10

0 Day 191 Ave= 8.7 le-03

-o. 10 Cluster 3-- 4 m

0 10 20 30 40 J 50 SO min g Illll Ill gllll Ill ! IWg Ill TEE Iflltlllllllll !l! IIII fVllfJ

0.oo3 I• 0 Day 189

'"'[ •' Ave= 2.86e-04 -0.oo3

0.03

0 Day 180 Ave-- 2.62e-03

-0.03

0.10

0 Day 191 Ave= 11.5e-03

-0.10

Cluster 4-- 7 m

Fig. 6. One-hour time series of w'T' at two depths on three different days: (a) 4 m; (b) 7 m. Note the 33-fold difference in scales at the left. Average values listed at the right are shown by dashed lines.

w'T'/tr origin provides a reasonably good fit to the average pdf of all six segments, as is shown in Figure 7 c. The integral of a function pieced together from the two segments is found by quadrature to be 1.00. Similarly, the first and second moments (mean and variance) found using quadrature are

0.31 and 1.10, respectively, which agree reasonably well with the observed values of 0.32 and 1.0.

The relatively large skewness, related to the difference in the coefficients of the negative and positive exponentials in Figure 7c, indicates the efficiency with which heat is dis- persed as the bigger eddies overturn; one would expect, for example, small skewness in data from a nonbreaking internal wave field in the thermocline. Large kurtosis reinforces the qualitative view that the most of the actual heat transfer takes place during intermittent events (i.e., w'T' spends most of its time near zero), and emphasizes the fact that there will be much variability in short-term estimates of heat flux.

In addition to showing basic characteristics of the process, results here confirm that the TC system is well suited to measuring oceanic heat flux. Its main limitation, as with any mechanical system, is the finite size of the apparatus used to mount the sensors. In the present configuration, the three rotors and TC pair are distributed in a horizontal plane across a span of about 40 cm. We have seen that the eddies where most of the covariance between vertical velocity and temperature resides have time scales of at least a minute, which in a 10 cm s- • flow means a horizontal length scale in excess of 6 m, an order of magnitude larger than the scale of the clusters.

3. HEAT FLUX IN THE EKMAN LAYER, CEAREX 89

During the late winter CEAREX 89 O Camp drift, we encountered strong diurnal tidal forcing north of a bathymet- tic rise in Fram Strait named Yermak Plateau, which tended to keep the mixed layer deep and more or less continuously stirred [McPhee, 1991]. As part of the O Camp turbulence experiment, several TCs were mounted on a rigid mast spanning up to 11 m, which could be lowered as a unit to depths in excess of 100 m. The mobile instrument frame (MIF), shown schematically in Figure 8, was intended primarily for investigating internal waves and turbulence in the lower mixed layer and upper pycnocline but could also be positioned higher in the boundary layer to serve as an extension of the fixed masts suspended from the ice. The clusters are carefully aligned on rigid stainless rods, with orientation and depth determined by compass, tilt meters, and a pressure gauge in the unit. In this section, measurements of turbulent stress and heat flux from one complete diurnal cycle, with the MIF positioned about a third of the way down in the mixed layer, are presented. Discussion here

TABLE 2. The First Four Moments of the w'T' Time Series Along With the Ratio of the Mean to the Sample Standard Deviation rr for Six l-Hour Runs

(w' T'), tr 2 , Depth, K-cm s - • (K-cm s - • ) 2

Day m (x 103 ) ( x 104) Skewness Kurtosis (w'T')/tr

189 4 0.53 0.20 1.27 4.39 0.37 7 0.29 0.13 1.17 7.30 0.25

180 4 2.05 2.89 0.67 1.99 0.38 7 2.62 7.02 1.65 5.88 0.31

191 4 8.71 65.21 1.57 6.26 0.34 7 11.47 183.09 0.55 2.72 0.27

If the probability distributions were Gaussian (normal) the skewness and kurtosis would be zero, with approximate standard deviations of the estimators of 0.04 and 0.08, respectively [Press et al., 1988]


1.2

0.8

0.4

0.0

(a)

Cluster 3• 4 m

ß Day 189

+ Day 180

o Day 191

0.36

2

(B)

Cluster 4• 7 rn o

ß Day 189

+Day 180

•! o Day 191 •(w'r'/o) = 0.28

,

..• --•,,

I I • • ---• 0

w'T'!o

o., p = e-2-411w'T'/o! 0.6

0.4

0.2

I I I

0.0 -2 0 2 4 w'T'/ o

Fig. 7. Probability distribution functions constructed by smoothing 50-bin histograms from three 1-hour time series of w'T' at levels 4 m (Figure 7a) and 7 m (Figure 7b) beneath the ice during the MIZEX 84 drift. Mean values are indicated by the dashed lines. In Figure 7c a composite pdf from all six time series is shown, along with an idealized pdf made up of two exponential segments.

•• Wire rope to small winch at surface, and condueling • cablCworksm SBtafi2-11 deck unit, interfaced to aSun

• TC1 Aluminum fins orient clusters into flow, so [J .•l•q that all three mlhogonal components see

• roughly the same magnitude • TC 2 I Slainless mast sections, 2 m long, are joined

• with fiat plates at each end, machined to

I I maintain alignment over the entire masC TC3

TC4

Sea-Bird Electronics SBE 9 Underwater

Unit (modified): Digiquartz • sens• Digicourse compass Spectron tilt sensors 12 SBE T/C frequency channels 18 Smith rotor em freq channels

5

Ballast

Fig. 8. Schematic of the mobile instrument frame (MIF) as used in CEAREX 89 O Camp. The whole structure can be lowered to depths in excess of 100 m.

Each turbulence cluster crc) comprises three small, partially ducted rotors approx. 4 cm in diameter, mounted in a horizontal plane along three orthogonal axes, near Sea-Bird temperature and conductivity sensors. Each sensor is sampled 6 times per second

is limited to four turbulence clusters spanning 7 m in the vertical, because of an electrical problem in one cluster and a malfunctioning thermometer in a second.

General conditions during a tidal cycle from day 103.2 to 104.2 (where 103.5 is 1200 UT on April 13, 1989) are shown in Figure 9. A steady, easterly wind drove ice westward at about the customary 2% of wind speed, but the well-behaved ice drift masks a fairly dramatic reversal of upper ocean currents and oceanic boundary layer stress. Apparently, the pack ice was responding more or less rigidly to the wind forcing over a much larger scale than the diurnal tidal currents, and internal ice stress gradients compensated for the change in water stress locally. The mixed layer depth (defined as the depth at which tr t exceeds its surface value by 0.01 kg m -3) varied between 70 and 100 m over the day [Padman and Dillon, 1991].

After screening to ensure that all velocity components exceeded 1 cm s-l, 1-s averages of u, v, w, T, and C at each cluster were treated by grouping the sums and product sums into 15-min "flow realizations," then calculating the mean quantities, heat flux vector, and Reynolds stress tensor.

MCPHEE: TURBULENT HEAT FLUX IN THE UPPER OCEAN UNDER SEA ICE 53'71

Wind/50

Ice

U(2)

I ..... I ..... ! ..... I ..... I

103.200 103.450 103.700 103.950 104.200

Fig. 9. Hourly vectors of surface wind (divided by 50), ice drift from satellite navigation, and 15-min samples of water velocity measured in a reference frame drifting with the ice at 2 and 15 m below the ice-ocean interface. Direction is indicated at left, so that the mean wind and ice vectors are almost due west.

These data were further smoothed with a 5-point running mean. Friction velocity, defined as a vector •, -- •'/r J/2, where •- • (u'w') + i(v'w') is the horizontal Reynolds stress, is shown for selected levels (including one on the fixed instrument frame) in Figure 10. Note that to facilitate comparison between Figures 9 and 10, the vectors are aligned differently' if the vectors line up, h, is actually 45 ø clockwise from relative velocity.

Water temperature and vertical turbulent heat flux, averaged across the MIF, are shown in Figure 11, with error bars indicating the magnitude of the standard deviation among the four clusters. The heat flux measurements are consistent

from level to level and show a bimodal structure with

minimum flux coinciding with currents from the northwest There is a mean upward flux of about 12.5 W m -2 and a significant period of negative flux beginning shortly before 103.4. While the curves are not exactly sinusoidal, it is clear that temperature lags heat flux by roughly a quarter cycle, indicating flux divergence.

What drives the negative flux is not obvious. In general terms, there is a source of heat in the thermocline at the base of the mixed layer and a sink at the surface from loss of

"•/••'•" 15m

I , , , , , I , , , , , I , , , , , I , , , , , I

103.2ffi 103.450 103.7ffi 103.950 1 •.2ffi

Fig. 10. Measurements of vector friction velocity h, at three levels: 8 m below the ice on the fixed instrument frame, and 15 and 22 m on the MIF. The vector time series have been rotated 45 ø

counterclockwise with respect to those of Figure 9. Note that sample times do •ot overlap precisely from the fixed to mobile frames.

-1.82 , , , Teml•,rature ,

................... 4 -1.85 • • • '

103.2 103.3 103.4 103.5 1•.6 103.7 103.8 103.9 1• 1•.1 1•.2

Turbulent Heat Flux

= 0 ............. ...... ............................. ..,-]

103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9 104 104.1 104.2

Day of 1989

Fig. 11. (a) Mean temperature across four clusters spanning 7 m on the MIF. Error bars represent 1 standard deviation of the measurements. (b) Turbulent heat flux, pcv (w' T'), averaged across the frame, with error bars indicating 1 standard deviation of the four clusters.

sensible heat in open leads and from conduction through thin ice, plus the possibility of melting at the base of thick ice. Thus the external driving suggests a continuous upward flux, and in fact, the near-surface TCs on the fixed frame rarely showed negative values. One can easily imagine a system which advects a mixed layer with a positive north-to-south temperature gradient back and forth across the measurement site. If the mixed layer were always in local equilibrium with the ice cover, there would be a sinusoidal variation in heat flux, but it would always be positive, and it would be in phase with the elevation of mixed-layer temperature above freezing. The actual situation may arise from preferential advection of colder water from the north at middepth in the boundary layer during the southward part of the diurnal cycle, and vice versa. As is shown below, there is appreciable shear in the mixed layer; thus it is possible for cold water to underrun near-surface water that has warmed during the previous northward excursion, leading to a positive temperature gradient favorable for downward flux. It is noteworthy that in this case the ice is constrained by internal forces to drift westward despite the large north-south current reversals. If instead it were free to advect with the currents, the heat flux picture would probably be quite different.

Barring radiative flux divergence or phase change, the heat conservation equation may be written

--= (w'r') -- •7h'F q ot oz

where the last term on the right is the horizontal divergence of the combined advective flux and horizontal turbulent flux.

The temperature change with respect to some starting reference will thus be the time integral of the right-hand side, the first term of which can be estimated by numerical integration of the vertical turbulent flux gradient time series. Figure 12 shows the calculation, with the gradient for each sample obtained from the slope of a least squares linear fit of heat flux from the four clusters. The results indicate that advec-

tion is dominant in the local temperature change, but that vertical flux divergence also plays an appreciable role.


xlO -3 Change in Temperature 4

2

0

-2 Flux Divergence

-10

-12

-14

-16 ' '

Day of 1989

Fig. 12. Change in frame-average temperature from its initial value (unmarked curve). Numerical integration with respect to time of the vertical temperature flux divergence, -O(w'T')/Oz (curve marked with asterisks).

One would hope to estimate the turbulent eddy diffusivity by relating the measured heat flux to the temperature gradient in the mixed layer; however, the gradient is very small. From general scaling principles, during times of maximum flux we expect a change in temperature across the four TCs of the order of 1 mK (millikelvin). The thermometer manu- facturer specifies a calibration accuracy of 10 mK, but the platinum resistor sensors can resolve much smaller differences in temperature; in fact, I have used them on occasion to trace the pressure dependence of the freezing temperature in the mixed layer, which is roughly 0.8 mK m -• . There is thus some hope that relative changes in temperature may be used to look at small gradients.

To proceed along these lines, I assumed that at first order, the net vertical temperature gradient over the entire tidal cycle was negligible compared with gradients associated with the diurnal signal. By adjusting each thermometer by a constant amount so that the mean gradient over the 1-day period vanishes, the systematic errors from constant calibration offsets are removed, along with compressibility (poten- tial temperature) effects. Vertical temperature structure across the frame can then be displayed by removing the frame average, and plotting the time series of temperature deviation (Figure 13). The results show differences in temperature consistent with the sense of the heat flux; i.e., the negative heat flux around 103.4 occurs with a positive temperature gradient and vice versa around 104.0. The maximum temperature difference is about 0.8 mK shortly after 104.0. Least squares linear fits to the temperature deviations versus vertical displacement furnish the temperature gradient, which is plotted along with the average heat flux for the two inner clusters in Figure 14. Generally, the two time series are negatively correlated with only a few instances of "countergradient" flux. No attempt has been made here to assign errors, because little is known about the response of individual thermometers at this resolution.

To provide a cross check on the eddy heat diffusivity, ! also compared current shear with turbulent stress. Neglect- ing small rotations across the frame, shear was calculated by linear least squares fits of the current speed versus vertical

xlO 4 Temperature Deviation 6

+-4 o--2

4 .--0m

[',. 24 ',.:, 2 / \ ii

s• ,.; \i :

" •f i : :' o•: ,. '"'," •' "', "' ","'"' -2 ¾'• ", f"'" •

-4

_6 • ! 104.1 104.2

Day of 1989

Fig. 13. Deviation of temperature at each cluster from the frame average, after the daily mean temperature of each cluster has been adjusted to remove any average gradient. The legend indicates vertical displacement in m from the lowest cluster. At time 103.45 the temperature at the uppermost cluster is about 0.6 mK greater than the lowest cluster, and at 104.05 it is about 0.8 mK cooler.

displacement (Figure 15a). Here one cannot eliminate systematic errors by assuming zero mean shear over the tidal cycle; however, a series of tow tank tests have indicated errors of 3% of current speed for individual current meters within the clusters [McPhee, 1989], which provides a con- servative error estimate for speed. The error bars for shear in Figure 15 are the square root of the expected variance in slope, following Press et al. [1988]. The average magnitude of the turbulent stress (u,•) across the frame is also shown in Figure 15, with error bars equal to the standard deviation.

A time series of eddy viscosity, K m = lt.2/(OU/Oz), may be constructed from the information in Figure 15 and is shown in Figure 16a. Errors have been propagated by using a Taylor's series expansion for the denominator dropping all but first-order terms. Also shown is the eddy heat diffusivity, Kr = H/[pcp(OT/Oz)], for samples where the magnitude of

xlO 4 Temperature Gradient

oOOOoO o o o o oo • ø o oo o ø oO / n L_*____o_ •. o_"_ .............. _'Y_ .*,,. m*_* ............................................ •___]

ß , / • o o • o• i•o• o,.o I

/ o oO.- oø ø '/ / %,,o ooo o /

103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9 104 104.1 104.2

Turbulent Heat Flux 40 , , , ,, ,

, *, ***** ***** B

eq 20 **** * , *** o __:::_:':X., ........... t'- ..... ': ....... :::; ..................................... -.• i I ****** i i i i i i

103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9 104 104.1 104.2

Day of 1989

Fig. 14. (a) Temperature gradient from the slope of a least squares fit to the temperature deviations of Figure 13. (b) Frame average turbulent heat flux, from Figure 1 lb.


i x10 -3 .... S.h• .... A

103.2 103.3 103.4 103.5 103.6 103.7 103.8 103.9 104 104.1 104.2

3 x10'4 , , , Kinematic Turbulent Stress B

01 i i i i t 103.2 103.3 103.4 103.5 103.6 103.7 10•.8 103.9 104 104.1 104.2 Day of 1989

Fig. 15. (a) Velocity shear across the MIF from the slope of a least squares fit of relative speed. Error bars signify the square root of the expected variance in the slope, assuming an error of +-3% of current speed. (b) Turbulent stress u,: averaged over four clusters. Error bars are I standard deviation.

the temperature gradient exceeded 0.05 mK m -• . While the calculation of temperature gradient is questionable in the sense that the thermometers have been pushed far beyond their stated accuracy, the overall similarity of eddy viscosity and diffusivity lends credence to the method.

According to Rossby similarity, eddy viscosity at a particular nondimensional level in the flow should vary as u,:/œ. Nondimensionalizing the eddy viscosity, K, = JK,,/u, 2, smooths out some of the variability in K m (Figure 16b), but much still remains, especially during the hours just before midnight on day 104 and at the start of the record. While the source of the variability in eddy viscosity for similar stress conditions is not the main focus of this work, there are several possible explanations, among them (1) changes in stability of the water column, (2) increased levels of turbulence in the boundary layer due to internal wave activity

0.2 . . . Eddy. V'mcosi•., •ivi• . .

2 2

near the base of the mixed layer, and (3) local topographic effects from pressure ridge keels.

I investigated the stability question by calculating a time series of the salinity gradient across the frame in the same way as for temperature. The results, though somewhat inconclusive, suggest a slightly stable density gradient at around 103.4. However, the minimum Obukhov length associated with it is about twice the mixed layer depth, and it is unlikely to have had much discernible effect on the turbulence scales.

From Figure 16b there appears to be a minimum nondimensional eddy viscosity "floor" around 0.05, which is about 3 times the value inferred for the maximum K, in the boundary layer under the 1972 AIDJEX ice station, based on spectral peaks found in a similar ice-ocean turbulence experiment [McPhee and Smith, 1976], and corroborated by subsequent model/measurement comparisons [McPhee, 1987; McPhee eta!., 1987; McPhee and Kantha, 1989]. Using a turbulence model derived from the earlier studies, I inferred from the mean flow and Reynolds stress behavior under O Camp [McPhee, 1991] that boundary layer turbulence was being enhanced by some agent (probably breaking internal waves near the base of the mixed layer) in addition to shear at the ice-ocean interface. The direct eddy viscosity measurements here tend to reinforce that view.

The still greater increase in mixing length scale at the start of the period and again between 103.9 and 104.0 may result from a local topographic effect. At those times, relative flow was from the southwest and south, across a rough, first-year pressure ridge keel approximately 60 m away. An alternative explanation may be that dramatic "bore" events observed in the upper pycnocline contribute to a periodic increase in boundary layer turbulence levels.

The mean heat flux in the mixed layer, 12.5 W m -2, compares with an average heat flux out of the core of the Atlantic Water (above a temperature maximum around 200 m) of about 25 W m -2 estimated from microstructure profiles over the Yermak slope (from day 103.0 to 110.0) by Padman and Dillon [1991], using an eddy diffusivity based on the critical flux Richardson number, the local buoyancy frequency, and the dissipation rate. While they did not estimate heat flux in the mixed layer because of the small temperature gradients, their mean value decreased to about 5 W m -2 near the top of the pycnocline around 80 m (compare their Figure 6). The direct flux measurements made from TCs 4 and 8 m below the ice on the fixed mast

near the surface averaged about 9 W m -2 over the period 103.0 to 110.0; however, the sampling tends to emphasize periods of higher stress and may not be directly comparable.

. Nomlin]imsl, 'o•, Eddy

0.15 +

0'1 f 4.

o.o '+.++ 0 I I I t I

Day of 1989

Fig. 16. (a) Turbulent eddy viscosity (open circles with error bars), and turbulent heat diffusivity when the temperature gradient exceeded 0.05 mK m-I (asterisks connected by solid lines). The two measures are independent except for the contribution of w' to the fluxes. (b) Eddy viscosity nondimensionalized by ,,2/f.

4'+

4. 4.

++ + ++ + 4. 4-+

+4+ +

4. ICE-OCEAN HEAT FLUX

By regulating the temperature at the interface to be always near freezing, sea ice provides a rigid constraint on heat flux in the upper part of the oceanic boundary layer. The main elements of enthalpy conservation at the ice-ocean boundary are conduction of heat through the ice balanced by latent heat changes as freezing or thawing occurs at the interface, plus upward heat flux from the ocean. In this section we .consider measurements of turbulent stress and heat flux in

the upper ocean, along with mixed layer temperature and salinity, from three different projects. The aim is to evaluate parameters in simple models for heat and mass transfer at the ice-ocean interface.

5374 MCPHEE' TURBULENT HEAT FLUX IN THE UPPER OCEAN UNDER SEA ICE

The oceanic heat flux at the lower ice surface, H =

pcp(w'T')o, depends primarily on AT = Tml - T 0, the difference in temperature between the undisturbed fluid at the far extent of the boundary layer and the interface, and on turbulent mixing in the boundary layer, characterized by friction speed at the interface, u, o. It is convenient to group the three parameters as a number indicating the nondimensional change in temperature across the boundary layer:

AT

ePT = (w'T')o/U,o •T is the reciprocal of a heat transfer coefficient (i.e., Stanton number) based on the friction velocity instead of the far-field velocity.

Here we consider two approaches to specifying the form of ß T. The first is to simply assume that A T is the difference between the mixed layer temperature and its freezing temperature and that the Stanton number for a particular site is constant, i.e.,

(w'T')o = Chtt,0(Tml- Tf)

Using a constant Stanton number, while attractive from a computational standpoint, neglects important issues of heat and mass transfer across a phase change boundary. An extensive data base from engineering studies of turbulent scalar flux across a hydraulically rough surface shows that the Stanton number has a Reynolds number dependence, where the Reynolds number is the ratio of inertial to viscous forces in the momentum budget [e.g., lncropera and De Witt, 1985]. Thus •T may depend in some way on u, o. In addition, sea ice rejects most of the salts as it freezes (a typical salinity for newly formed ice is 5-7 psu); thus melting can have a significant effect on heat transfer in two ways: first, by elevating the freezing temperature at the interface relative to Tf (because the salinity is less there than in the mixed layer proper), and second, by stabilizing the turbulent boundary layer so that the efficiency of turbulent exchange is diminished. These concerns are addressed in the second

model, developed by McPhee et al. [1987], based on the work of Yagiom and Kader [ 1974] (hereinafter referred to as YK), and Owen and Thomson [1963]. They divided the boundary layer conceptually into a region where molecular viscosity and diffusivity is important, called the transition sublayer, embedded within the fully turbulent boundary layer. McPhee et al. [1987] simplified the YK formula for the inverse Stanton number across the transition sublayer, and expressed the total nondimensional changes in temperature and salinity as

(I) T = (I)turb + bRe I/2pr2/3

u,0(Sml- S0)

(w'S')o = (I)turb + bRe I/2Sc2/3

where Pr = dar, Sc = das, with v, aT, and as being the kinematic molecular viscosity, heat diffusivity, and salt diffusivity, respectively. The Reynolds number is Re = u.ohts/V, where hts is the thickness of the transition sublayer. In the YK theory it is taken to be the scale of the roughness elements: hts = 30z0. The contribution of the fully turbulent part is based on a similarity theory for the

planetary boundary layer under the influence of surface buoyancy flux [McPhee, 1983], given by

I //,O•:NI/, 1 1 = + •t.-•, •log fzo 2•Nn. t:

where k is von Kfirmfin's constant, SON is a constant, approximately equal to 0.05, and r/. is a buoyancy factor, equal to 1 when stratification is neutrally stable. For seawater near freezing, the turbulent contribution is generally small relative to the change across the transition sublayer, especially for salinity; nevertheless, (I)turb has a potentially important feedback via the buoyancy dependence when melt rates are extreme.

McPhee et al. [1987] also used hts = 30Z0 for the length scale in Re, and found from a limited sample that our measurements were best described when the constant b was

about half the value found by YK from laboratory studies. We noted at the time that z0 was relatively large for the MIZEX site, probably reflecting large, irregularly spaced roughness elements, and questioned whether it was the appropriate scale for the heat and mass transfer sublayer. In what follows, hts is treated as an independent parameter with b equal to a constant, 0.6, taken as an approximation to the YK laboratory result. The analysis below indicates that hts is not necessarily related to z0.

Extensive data sets including momentum and heat flux measurements near the ice-ocean interface are available

from three drift station experiments: MIZEX 84 in the marginal ice zone of the Greenland Sea (section 2); CEAREX 88 during fall freeze-up in pack ice of the eastern Arctic north of Murmansk, and CEAREX 89 O Camp during late winter in pack ice north of the Yermak Plateau and Fram Strait. The strategy is to consider each experiment sepa- rately, classifying measured heat flux according to friction velocity and Tml - Tf, then using the resulting distributions to estimate c h and hts for each site.

Examples in section 2 demonstrated that the turbulent heat transfer process consists of a series of large negative and positive excursions in w'T', sometimes lasting several minutes, and often an order of magnitude larger than the magnitude of the mean heat flux. An averaging time long compared with the time scale of the individual eddies is thus required for the covariance statistics to "settle down." On the other hand, sea ice drift is only occasionally steady for hours at a time, so a compromise between sampling the turbulent field and other, lower wavenumber phenomena is required. Past experience has suggested that averaging the turbulence statistics over 15-min "realizations," then smoothing or averaging these samples over time periods of an hour or more provides a consistent view of the turbulent field, as demonstrated in section 3. The approach taken here was as follows: (1) Calculate the turbulent statistics for every 15-min segment of data meeting minimum mean velocity criteria for each component; (2) group the 15-min realizations into 3-hour bins, calling a 3-hour average valid if it included at least six realizations (90 min of data); (3) classify the average heat flux from each 3-hour average according to Tml - Tf and u,0, where the friction velocity is adjusted to the interface assuming an exponential decay of stress with depth (Ekman-type model [see McPhee, 1990]); and finally (4) consider a u,0-AT bin valid if it contains at least three 3-hour averages, i.e., nominally 9 hours of turbulence data.


Only data from clusters between 2 and 8 m below the ice undersurface were considered, since deeper clusters may reflect changes in the mixed layer heat budget not necessarily related to the surface flux, as found in section 3.

Results for each of the drift experiments are shown in Figure 17. For the summer (MIZEX 84) and fall experiments (CEAREX 88), u,0 varies from 0 to 2 cm s -• and Tml - Tf varies from 0 to 0.32 K, with each axis divided into eight bins. During MIZEX 84 there were periods of higher temperature deviation (see, for example, the examples discussed by McPhee et al. [ 1987], and in section 2), but there were not enough data to satisfy criterion 4 above. During the last part of the MIZEX 84 drift, the boundary layer was stratified almost to the surface, and internal wave drag was a major part of the momentum balance [McPhee and Kantha, 1989]. This greatly complicates the heat flux determination, and those data were not used here. For the winter experiment, heat flux and temperature deviation were much reduced, so in Figure 17c, the total temperature deviation range is reduced to 0.05 K and the axes are divided into five bins

each. Note that nearly all of the samples from CEAREX 89 would fall in the first temperature bin of the other two experiments.

While there are a few inconsistencies, the general trend of heat flux increasing with both increasing temperature deviation and increasing friction velocity is evident. Figure 17a and 17b are directly comparable, and where there is overlap between U,o-/XT bins, the agreement is relatively good. It is not obvious that this should be so, because the under-ice boundary layer roughness was considerably larger during MIZEX 84 than during CEAREX 88.

Differences among the three sites may be quantified by calculating minimum least squares error estimates of the parameters Ch and hts for the two heat flux models described above. For the transition sublayer model, this involves solving the equations for each bin over a range of values for hts, choosing the one that minimizes the error between modeled and measured heat flux. It was assumed that

melting at the ice undersurface absorbed all of the oceanic heat for both MIZEX 84 [McPhee et al., 1987] and CEAREX 88 [Wenlaufer, 1991] where temperature gradients within the ice were small. No ice temperatures were measured in CEAREX 89, so conductive heat flux was estimated to be between 5 and 15 W m -2, following Maykut's [1986] recom- mendation for perennial ice in late winter/spring with a modest snow cover. Table 3 lists the results, with the range for CEAREX 89 reflecting the range in assumed heat loss through the ice. Also listed is under-ice surface roughness, computed from all of the valid turbulent flow realizations from clusters 4 m below the ice at each site, according to

z0 = (4.0 m) exp (-(kU4/u,o))

where U4 is the mean speed relative to the ice, and the angle brackets denote the average over all 15-min realizations. There is clearly much less site-to-site variation in h ts than in z0. Despite almost a factor of 2 difference in the oceanic drag coefficient (referenced to relative current at 4 m) between MIZEX 84 and CEAREX 88, the heat flux parameters are very similar.

Figure 18 shows the heat flux during CEAREX 88 from three sources. Three-hour averages of direct flux measurements 4 m below the ice have been interpolated to a regular time grid and smoothed with a 24-hour running mean. The

heat flux calculated from u,0, T4, and S 4 according to the modified YK theory with hts= 0.09 m is shown by the dashed curve, where u, o is determined by adjusting the smoothed stress measurements at 4 m to the interface.

Finally, the curve marked with asterisks is the average oceanic heat flux inferred from bottom melting and temperature profiles at 11 instrumented sites in the ice floe from which the instrument frame was suspended [Wettlaufer, 1991]. Individual time series from Wettlaufer's study show much variability reflecting large differences in ice ablation at the 11 sites, and it is natural to question whether turbulent heat flux measured with one instrument cluster would be any more representative of the average than heat flux estimated from ice ablation at one site. While the data here do not

answer the question definitively, two comments are in order. First, by making the turbulence measurements beyond the surface layer (i.e., the layer near the interface where turbulent stress and mean shear depend on the distance from the interface and thus on z0), the dependence on purely local conditions is lessened greatly. To use an atmospheric analogy, the instrument level of 4 m in the under-ice boundary layer corresponds roughly to measuring turbulence at the top of a 120 m tower in the atmosphere. An observer on the ground might measure quite different values of heat flux over local terrain changes near the tower (e.g., a highway versus the adjacent field), but intuitively, we would expect measurements at the top of the tower to represent an average over a fairly broad area. The second comment is that mean temperature and salinity at 4 m are most likely representative of mixed-layer properties. Consequently, the correspondence of the model and smoothed turbulence data over much

of the time series suggests that local effects are not of overriding importance.

Since heat transfer is apparently insensitive to the turbulent Reynolds number formed using the momentum surface roughness z0; and since the bulk transfer coefficient C h shows little variation from site to site, an approximation to the heat and mass transfer model that is much simpler than the YK approach may be appropriate, at least for the range of values covered by the measurements presented here. Define turbulent temperature and salinity scales as T, = (w'T')o/U, o and S, = (w'S')o/U, o, respectively. As melting or freezing occurs, the elevation of the interface will adjust isostatically as the mass of the ice changes, with velocity w = w 0 + w i where w 0 is the component due to freezing or melting and w i is the percolation velocity arising, for example, from melting at the upper surface or within the ice [McPhee, 1990]. (The interface velocity w is generally so much smaller than U,o as to be inconsequential for the boundary layer dynamics [Melior et al., 1986].) The heat equation is dominated by the phase change at the interface and conduction of heat through the ice and may be expressed by the balance

T,u,o = woQL + it

where//is heat conduction in the ice divided by the product of density and specific heat of seawater, and QL is latent heat of fusion (adjusted for brine volume) divided by specific heat. A typical value of Q c for ice of salinity 4 psu and temperature -1.8øC is 74 K [Maykut, 1985]. If we assume there is no appreciable salt accumulation in a control volume surrounding the interface, the balance is advective, with the turbulent flux from below equal to the advection out of the


A

I MIZEX 84: 2, 4, 7, 2r m I

• 9 o

I CEAREX 88: 2, 4, 8 rn ]

• 9o

Fig. 17. Perspective views of turbulent heat flux as a function of u,0 and AT = Tml - Tf. Each axis has been divided into bins for classification of all 3-hour averages of heat flux. Binned averages are shown only if there are at least three 3-hour samples. Data are shown for near-surface clusters in (a) MIZEX 84 during summer in the marginal ice zone of the Greenland Sea, (b) CEAREX 88 in eastern Arctic pack ice north of Murmansk during fall freeze-up, and (c) CEAREX 89 north of Fram Strait in late winter. Note the change in scale for both heat flux and AT in Figure 17c.


2.5

I CEAREX 89: 2, 4, 8 rn ]

2.0

Fig. 17. (continued)

control volume less advection in: S,u,o = wAS, where AS is the difference in salinity between water leaving the control volume and ice entering. Substituting the expression for w 0, we have

S,= F --- AS u,oQl• U,o/

Using the bulk coefficient Ch, the expression for the turbulent scale temperature is simply

T, = chAT

ancl for D•rh•lont qc',qlo qalinitv

S,- Qz. ,,•-Qt.' AS In a bulk formulation as before, AT = Tml - Tf and AS = S ml - Sic e where S ml is the mixed layer salinity and Sic e is ice salinity. We thus have a very simple expression for the salinity flux at the interface (especially if there is no percolation velocity or heat conduction, so that the second term in the S, expression is zero) which is computationally efficient compared with the modified YK formulation. The present results indicate a value around 0.006 for C h. The importance of salinity flux is that it largely determines the surface buoyancy flux at low temperatures and will thus have a

relatively greater impact than heat flux on polar mixed layer dynamics.

The main drawback to this simplified model is that if AT and AS are considered to be the actual change in temperature and salinity across the boundary layer, then they are

equal to Tml - Tf and S ml - Sice only if there is no melting or freezing (w0 = 0). Otherwise, they are AT = Tm! - T O and AS = S0 - Sice, where To and S0 are temperature and salinity at the interface, constrained to be at the freezing point. Both will vary with the melt rate. The modified YK model treats this explicitly and in that sense seems more realistic. It also provides, for example, a mechanism for supercooling and frazil vroduction associated with differing

TABLE 3. Comparison of the Underside Surface Roughness Calculated From Stress and Velocity at 4 m, the Thickness

of the Transition Sublayer in theModified Theory of Yaglom and Kader [1974], and the Heat Exchange

Coefficient From Three Drifting Ice Stations

Drift Station z0, m h ts, m C h

MIZEX 84 0.091 0.10 0.0060 CEAREX 88 0.021 0.09 0.0058 CEAREX 89 0.062 0.06-0.11 0.0050

The range in h ts for CEAREX 89 reflects a range in assumed conductive heat flux through the ice of 5 to 15 W m -2


,••, ß 3-hr data • Smooths! Data

..... Modeled Data

• " i• ;, . . : ß • 30 " ' " i a• " ß • ' . ; ß

ß ', ß ß

0 ß ß

Day of 1988

Fig. 18. Oceanic heat flux during CEAREX 88. The plain solid curve is the interpolated 3-hour data from the cluster at 4 m, smoothed with a 24-hour running mean. The dashed curve is heat flux modeled using temperature, salinity, and turbulent stress at 4 m, with fits = 0.09 m. The solid curved marked with asterisks is mean heat flux inferred from temperature and ablation measurements at 11 sites in the ice floe, reported by Wettlaufer [1991].

molecular diffusivities. On the other hand, the data in this section do not clearly show that the YK formulation is superior. It seems that controlled laboratory studies with ice in salt water might be quite helpful in determining whether the more sophisticated treatment is necessary.

5. SUMMARY

Oceanic turbulent heat flux measurements from three

drifting ice station experiments suggest the following: 1. Turbulent heat flux occurs predominantly in eddies

with time and length scales similar to those for turbulent momentum transfer. Power spectra of vertical velocity and temperature deviations are similar, when normalized by the variance, with each having an extensive -5/3 slope region in the log-log representation. There is a broad peak in squared coherency between w' and T' centered at a frequency around 1/(100 s), equivalent to a horizontal wavenumber of about 1/(13 m), assuming that eddies advect past the instru- ments at the mean flow speed. Power spectra of the w'T' time series also show an extensive -5/3 region, with little coherency between clusters mounted 3 m apart vertically, suggesting that the eddies responsible for heat flux have a relatively small vertical to horizontal aspect ratio. The w' T' signals show a series of positive and negative events typically lasting a minute or more, with peaks often an order of magnitude greater than the mean value. As one might expect from the steep spectral fall-off, trials with various length low-pass filters showed that most of the turbulent heat flux can be captured with fairly coarse time averaging (Figure 5); for example, even after a 30-s smoothing of w' and T', the zero-lag covariance still accounts for 80-90% of the heat flux calculated from 1-s samples. Self-similarity of the turbulent exchange process is demonstrated by the similarity of probability distribution functions estimated from histograms of the six w' T' series, normalized by the standard deviations. The pdf's should help in designing efficient sampling strate- gies, especially when power and data storage capacity are considerations.

2. In the outer part of the boundary layer, away from the immediate effects of the ice-ocean interface, there is reasonably close correspondence between eddy viscosity (the ratio of turbulent stress to shear in the mean flow) and eddy heat diffusivity (the ratio of turbulent heat flux to temperature gradient). At the CEAREX 89 site, the nondimensional base level of both exchange coefficients appeared to be about fK/u. 2 = 0.05, which is 2 to 3 times greater than the value inferred from other experiments in the central Arctic. The difference may be that in the previous experiments the boundary layer developed when wind forced the ice to move over relatively quiescent (slow moving) water. In CEAREX 89, the boundary layer most often occurred in response to energetic tidal flow relative to the ice, which was constrained by internal ice stress gradients from drifting with the tide. It is plausible that this active flow, which had a much more energetic internal wave field in the underlying pycnocline, enhanced turbulence by providing sources of turbulent kinetic energy apart from shear at the ice-ocean interface [McPhee, 1991].

3. Under a variety of heat flux conditions, heat exchange between the ice and ocean at three different sites was

unexpectedly uniform, as characterized either by a direct heat exchange coefficient c h or by the inferred thickness of the transition sublayer, hts (Table 3). This implies that under-ice surface roughness has a different meaning for the scalar fluxes of heat and salt than for momentum flux. In

physical terms, this probably means that the thin sublayers, across which molecular exchange dominates the shape of the scalar profiles, more or less follow the shape of the surface and are not much affected by the larger scale roughness elements that contribute substantially to the overall drag. In other words, heat and salt flux respond to the actual levels of turbulent stress in the boundary layer, as opposed to the speed of ice drift relative to the underlying ocean. These results do not imply that the Yaglom-Kader approach is inappropriate for sea ice; in fact, Figure 18 shows close agreement between the modified YK model and smoothed observations. They do, however, show that hts is not proportional to the momentum roughness length, z0.

4. Observations that the bulk heat transfer coefficient is

relatively uniform over a wide range of heat flux and under-ice roughness provide justification for developing a much simplified approximation for heat and mass transfer at the ice-ocean interface. While the approach neglects potentially important effects like the elevation or depression of freezing temperature at the interface, it furnishes an explicit expression for salinity flux in terms of the bulk heat transfer coefficient c h; the percolation velocity w i; conduction of heat through the ice, 0; the mixed layer temperature and salinity; and the friction velocity at the interface.

Acknowledgments. Funding for this work by the Office of Naval Research, Arctic Programs, through contract N00014-84-C-0028 is gratefully acknowledged. L. Kantha provided helpful comment and discussion.

REFERENCES

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M. G. McPhee, McPhee Research Company, 450 Clover Springs Road, Naches, WA 98937.

(Received September 12, 1991; revised January 6, 1992;

accepted January 23, 1992.)

mcphee 1992

Documents

turbulent heat flux

measured heat flux

heat transfer coefficients

turbulent fluxes

iceocean interface

oceanic boundary layer

eddy heat diffusivity

turbulent spectrum