1. Background

Gascard and Clarke (1983) and Clarke and Gascard (1983) divided processes that drive deep convection into a hierarchy of scales, each of which is needed for deep convection to occur. These include the gyre scale of order 100 km, the mesoscale of order 10 km, and the plume scale of order 1 km. Each of these levels of organization will be discussed separately here.

c. Driving deep convection: Plume scale

Direct observations of deep convection on the plume scale (kilometer and smaller) are very limited. Theoretically (Maxworthy and Narimousa 1994; Jones and Marshall 1993), high surface heat flux causes the formation of a thin surface thermal boundary layer that feeds rapidly descending “plumes” of dense water. The upward return flow consists of broader regions of slowly ascending water. The aspect ratio of the horizontal and vertical extent of these plumes of downwelling water is believed to be about 1. Predictions of the magnitude of vertical velocity can then be made by scaling arguments. With a depth scale H [which can be taken to be a deepening mixed layer depth Fernando et al. (1991)] and a surface buoyancy forcing B_o a velocity scale can be created on dimensional grounds:

w_non−rotaB_oH^1/3(1)

where a is a coefficient of proportionality.

If the timescale, H/w, exceeds the Coriolis timescale, 1/f, then the above parameterization breaks down as vertical motions are impeded by the effects of rotation. We can rewrite this equation to show that this transition occurs when

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where Ro* is the natural Rossby number, B^1/2_o/(f^3/2H). When Ro* is very small, timescales to transit the mixed layer are large enough to force a rotating scaling for the vertical velocity to become more appropriate:

w_rotbB_of^1/2(3)

where b is a coefficient of proportionality.

Both numerical and lab studies (Maxworthy and Narimousa 1994; Jones and Marshall 1993; Coates and Ivey 1997; Fernando et al. 1991; Coates et al. 1996) have attempted to determine these coefficients and at what natural Rossby number the transition to full rotational control occurs. Coates and Ivey (1997) point out that the transition is smooth and asymptotic, but rotational control is complete when Ro* is less than 0.1, and Fernando et al. (1991) report that rotational effects become important for Ro* less than 1.5.

Few values of the coefficients exist in the literature. For their smallest rotation Fernando et al. (1991) report a value for a of 0.45, Coates and Ivey (1997) report an a of about 0.5, and Coates et al. (1996) report 0.3. We will estimate a value of 0.52 from oceanic data.

A few direct observations of the magnitude and extent of plumes exist. Gascard and Clarke (1983) report float observations in the Labrador Sea during the winter of 1975/76 that reveal floats with 1–2-km separation to have vastly different vertical velocities: one 0.09 m s⁻¹ and the other negligible.

Lilly et al. (1999) report Labrador Sea vertical plume velocities observed at a mooring location in the range of 0.04–0.08 m s⁻¹ and estimate horizontal extents of 200–1000 m. Since the noise threshold on these measurements is 0.02 m s⁻¹ (Lilly et al. 1999), only the highest-velocity events can be seen.

As on the mesoscale, interaction with horizontal gradients on the plume scale have been postulated to effect the evolution and behavior of convection. Horizontal gradients drive the upright plumes to slanting paths (Haine and Marshall 1998) that encourage lateral exchange.

2. Instrumentation and data processing

Deep Lagrangian floats (DLFs) are designed to follow water motions in three dimensions and thus be close to fully Lagrangian (see D'Asaro et al. 1996). This is accomplished by having the same buoyancy as the surrounding water and by having a high drag. The floats (see Fig. 2) have a specially designed aluminum hull that survives pressures up to 2000 db and has almost the thermal expansion rate of seawater and nearly the same compressibility. Final adjustments to the compressibility are made using a motor-driven piston capable of up to a 25 cm³ change in volume (∼1.6 kg m⁻³ change in density). A 1 m² cloth drogue provides drag, keeping the float in phase with the surrounding water. The drogue is initially folded, in a low drag configuration and is opened after an autoballasting routine is completed. The floats sample temperature and pressure at 5-min intervals, measuring temperature with millidegree accuracy and pressure to about a decibar. A RAFOS (Rossby et al. 1986) receiver allows determination of horizontal position at 4-h intervals. The mission length is approximately 2 months, after which time the floats surface by dropping a weight and transmit their data via the ARGOS satellite system.

The float buoyancy is determined by the difference between its density and that of the surrounding water. Density changes within the convective layer are 0.1 kg m⁻³ at most. This is partially compensated by the thermal expansion of the float, which is close to that of seawater, resulting in no more than 1 g of float buoyancy change. The compressibility of the float is actively adjusted using the piston to give a 1 g/1000 db stabilizing buoyancy change, that is, upward at depth, downward at the surface. This avoids the undesirable possibility of a destabilizing buoyancy gradient resulting from errors in our measurement of the hull compressibility. In addition, chemical reactions between the anodized aluminum of the hull and the seawater result in a slow increase of the float's weight. The hull is partially painted to minimize these. In the 1997 deployments, software schemes were used to compensate for this weight increase using the piston. These were improperly implemented, resulting in a slow increase in the float's upward buoyancy over time. Our best guess is that the 1997 floats were a few grams light on average, resulting in perhaps 5 mm s⁻¹ upward drift relative to the water. Harcourt et al. (2002) explores the consequences of this drift for the computation of turbulence statistics. The 1998 floats did not suffer from this error.

Meteorological data, the National Centers for Environmental Prediction (NCEP) reanalysis product, were obtained for the winters of 1996/97 and 1997/98. These data have been shown (Renfrew et al. 2002) to overestimate heat flux magnitudes, perhaps due to bulk formulae constants not designed for use in high latitudes. A “corrected” dataset, in good agreement with measurements made aboard the R/V Knorr during the winter of 1996/97, was provided by G. W. K. Moore (1999, personal communication). The NCEP data falls on a 1.875° longitude grid and 1.9° latitude grid with 6-h sampling. Float positions were interpolated to this 6-h temporal grid, and NCEP parameters were then interpolated to the float positions. These meteorological float values were averaged over all floats at each time step in the dataset. An rms vertical velocity for each point in the dataset during convection was calculated, binned into the same 6-h time steps as the meteorology, and averaged to allow comparison with the meteorology.

All error bars reported in this paper follow the “bootstrap” method for 95% confidence level presented in Efron and Tibshirani (1993).

3. Data

4. Estimating heat flux

One can estimate the net atmospheric heat flux by calculating the change in oceanic heat content and assuming that this heat content change has resulted from a heat flux to the atmosphere, a one-dimensional model. Such a one-dimensional approximation will be especially accurate when constructed using float data, since Lagrangian data minimizes the effects of horizontal advection.

Using two profiles of temperature, the heat content change (and thus heat flux) can be approximated by integrating the difference between the temperatures, θ₁ and θ₂, dividing by the time elapsed between the two profiles, Δt, and multiplying by the density and heat capacity of seawater:

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This is illustrated schematically in Fig. 8a as the area between the two profiles of temperature.

By plotting all DLF data in temperature–pressure pairs (Fig. 9), this method can be applied. In 1997 the floats sank to 1000 db for their autoballast cycle and thus recorded a profile of vertical temperature. When they entered the mixed layer 7 days later, it is clear from the temperature record that the bottom of the mixed layer (defined by a 0.2°C jump in temperature) had deepened from about 500 db to about 700 db, and that a significant amount of cooling occurred while the floats were below the mixed layer. Once the floats entered the mixed layer, only millidegree variations in temperature were revealed. The warming associated with the restratification at the end of the records is also clearly visible. Due to their buoyancy, the 1997 floats only sampled within the mixed layer and not the transition zone at its base, making it unclear below what depth the temperature remained unchanged between the autoballast cycle and the end of the float missions. This uncertainty in how to close off the temperature curves introduces a large range of heat flux estimates. Filling in with values from the deepest data (drawing straight vertical profiles deeper than the actual data) and assuming that the temperature deeper than 1100 db remained the same between 18 February and 10 March yields an estimated heat flux of 490 W m⁻²; carrying the same approximation to 1500 db [the deepest mixed layers observed by March 1997 Pickart et al. (1997)] yields an estimate of 585 W m⁻². A different deep extrapolation, extending the temperature/depth trend from 750–950 db deeper, will close the temperature curves at about 1200 db. This method yields a heat content change estimate of 450 W m⁻². This large uncertainty led to a programming change for the 1998 deployment: all floats performed a non-Lagrangian profile at the end of their missions to close off the heat content change. Repeat CTDs made at the beginning and ends of the float missions (as seen in Fig. 1) yield estimates of heat content change to be about 400 W m⁻² over this same period, in good agreement with the low end of the range of estimates here.

The 1998 floats (Fig. 9b) were deployed earlier in the year and found on their first downward trip (toward autoballasting) that the mixed layer base was at only about 200 db. When the floats returned from the autoballast cycle, they were too heavy to enter the mixed layer and thus did not resample the mixed layer base. Eventually entrained, the floats observed millidegree variability of temperature along individual mixed layer transits as in 1997. Because of the programmed (non-Lagrangian) profiles at the end of the float missions, the heat content estimate is closed off and a firmer estimate can be made: 140 ± 50 W m⁻² assumed lost to the atmosphere.

A second way to estimate heat content change is to separate the records into layers and then estimate slopes of temperature change with time within each layer, ∂θ_i/∂t. Combined with the depth of the layer, H_i, an estimate of heat content change within the layer (and thus net heat flux from the layer) can be made:

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This is illustrated schematically in Fig. 8b as the temperature trend with time within two layers. In order to obtain the total flux, the net fluxes from each bin must be summed.

Assuming that the heat content change within a layer is the result of a net heat flux and assuming all net heat fluxes are effects of transfers to the atmosphere, a total heat flux to the atmosphere may be estimated by summing over the layers:

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We use two layers: the initial mixed layer (layer₁) and water initially below the mixed layer but incorporated into the mixed layer during the records (layer₂). This separation allows estimation of the heat flux from entrained water and heat flux from cooling of the “original” mixed layer to be made. The temperature time series in Figs. 10 and 11 show approximate fit slopes to the temperature records for both layers. In 1997, layer₁ was defined as the surface to 650 db and layer₂ was defined as 650–1100 db. As with the profile heat flux estimate, the total heat flux from this temperature trend estimate is quite sensitive to which depth is chosen as the depth of no net heat flux. Choosing 1100 db for the limit in 1997 will yield a conservative total heat flux estimate. In 1998, as seen in Fig. 9, the temperature of water below 950 db remained unchanged, so layer₂ was defined as the range from 400 to 950 db, with layer₁ extending from the surface to 400 db.

In 1997 the majority of the heat content change occurred in water entrained into the mixed layer, with the net heat flux to the atmosphere from layer₁ (the “original” mixed layer) estimated at 175 W m⁻² and from the entrained water in layer₂ estimated to be 225 W m⁻², and a total heat flux of 400 W m⁻². Using 1500 db as a cutoff yields a total heat flux estimate of 590 W m⁻². Overall heat flux in 1998 was much lower. Heat flux from the entraining water in layer₂ was ∼160 W m⁻², again a larger magnitude than the flux from layer₁, estimated at 40 W m⁻². Surprisingly, the upper water actually warmed during this period; however, the total heat flux was still to the atmosphere. This somewhat counterintuitive result—that a warming mixed layer could still effect a heat flux to the atmosphere—inspires more detailed exploration.

5. Convective scaling

The predominant driver for mixed layer vertical motion is the atmosphere. Wind introduces energy mechanically, and buoyancy loss to the atmosphere can drive overturning. The relationship between vertical velocity observed by the floats and these atmospheric parameters will be explored in this section. Specifically, the buoyancy scalings, w_rms = a(B_oH)^1/3 (nonrotating) and w_rms = b(B_o/f)^1/2 (rotating) and their dependence on the natural Rossby number, Ro*, will be evaluated.

Average values for the NCEP atmospheric parameters, float-based observations of w_rms, mixed layer depth H, and the two types of float-based heat flux estimates are summarized in Table 1. These values emphasize the difference between the winters of 1996/97 and 1997/98, with much higher wind velocity and heat flux during the period of the float missions in 1997 than those occurring in 1998.

Average NCEP heat flux from the winter of 1997/98, 150 ± 30 W m⁻², agrees well with both types of float heat flux estimates, 140 ± 50 W m⁻² from the profile estimate and 130 ± 25 W m⁻² from the temperature trend estimate. This agreement does not hold for 1997. The NCEP heat flux for 1997 is estimated to be 270 ± 40 W m⁻². Float estimates for that year vary widely due to the uncertainty in what depth to close off the 1997 heat flux estimates. However, even the minimum value of heat flux (choosing the maximum depth the floats reached to be the deepest depth to witness a net heat flux), 400 W m⁻², is consistent with the 400 W m⁻² estimate from the repeat CTDs, and greatly exceeds the NCEP value for that year.

The ratios w_rms/(B_oH)^1/3 and w_rms/(B_o/f)^1/2 estimate the coefficients from the rotating and nonrotating scaling relations. These ratios reveal b = 1 within the error bars for both years for the rotationally controlled scaling of w_rms = b(B_o/f)^1/2. Marshall and Schott (1999) found velocity to more closely follow the nonrotating scaling with the coefficients set to 1; however, they used a maximum plume velocity instead of rms velocity. Coates and Ivey (1997) found a = 0.5 for the nonrotationally controlled scaling, w_rms = a(B_oH)^1/3 at oceanically relevant scales, in good agreement with the results here. Since the natural Rossby number for both years is equivalent within the error bars and since there are only two years as data points, attempting any sort of declaration about which scaling may be superior is unrealistic.

In order to utilize the entire range of meteorological forcing that occurred during the float missions and thus obtain more than two data points for evaluation of scaling superiority, the time series of the NCEP parameters and vertical velocity were compared. A lagged correlation (Fig. 14) analysis using 1-day running means of the NCEP and float values reveals a very strong peak in correlation between w_rms and heat flux at a 0.5-day lag in 1997 and 0.75 days in 1998 (atmosphere leading the ocean). Less prominent peaks at the same lags occur with the wind velocities. These values agree very well with estimates of overturning times for the two years, 0.6 and 0.8 days for 1997 and 1998, respectively. In the following analysis, lagged atmospheric values are compared with oceanic vertical velocity, each with a 1-day running mean applied.

Figure 15 shows scatterplots of vertical velocity, with (a) heat flux and (b) wind speed. There is more scatter in the wind data, which is to be expected with the higher noise in the wind data (Renfrew et al. 2002), but both show a linear relationship between the meteorological parameters and observed oceanic vertical velocity. Some differences between 1997 and 1998 could easily be due to systematic NCEP flux errors.

Buoyancy flux is seen to be the dominant driver of mixed layer vertical motions, overshadowing contributions made by the wind. This result may be due, at least in part, to the larger scatter in the NCEP wind data.

Figure 16 shows the time series of the NCEP meteorological terms and the rms vertical velocity with its error bars. In addition, scaled buoyancy flux and wind magnitude are shown, as well as the fit vertical velocity obtained with a linear combination of buoyancy flux (as in Table 3, line 3). General characteristics of vertical velocity for both years are reproduced with these linear parameters.

Results for combinations of this form are found in Table 3. The skill of the nonrotating forms (lines 3 and 4) are marginally better than the rotating forms (lines 5 and 6). As further evidence, in the two cases with both the rotational and nonrotational scalings (lines 1 and 2), the coefficient for the rotational scaling was essentially indistinguishable from zero and there was no additional skill through inclusion of a rotational term (lines 1 and 2 have no more skill than 3 and 4). There did seem to be a significant and nonzero value for w_o, the intercept allowed in the linear fit. This term could indicate a sink of energy out of the system, perhaps to lateral transfers. The magnitude, 0.005 m s⁻¹ (lines 1 and 3), would be consistent with approximately a 10% loss of energy for the overall average w_rms of 0.016 m s⁻¹. While the term appears significantly nonzero, models forcing its value to zero (lines 2 and 4) have indistinguishable levels of skill at reproducing the observations.

This analysis finds a nonrotational scaling of vertical velocity, (B_oH)^1/3, to be marginally superior to the rotational scaling, (B_o/f)^1/2, because there is a weak dependence of w_rms on H. Coefficients for fits with (B_oH)^1/3 alone and low rotation agree well with laboratory predictions.

The high level of skill attainable with these simple linear fits is similar to the correlation (0.89) reported by Renfrew et al. (2002) in their comparison of the NCEP model to actual heat flux observations made on the R/V Knorr during the winter of 1996/97. Simple models of vertical velocity constructed using only buoyancy fluxes without wind do a remarkably good job of matching observed vertical velocity, almost to the accuracy of the meteorological flux estimates themselves.

6. Conclusions

Lagrangian measurements of deep convection in the Labrador Sea revealed energetic vertical motions forced by surface cooling. Despite the different meteorological conditions of the two years, convection and a rapidly deepening mixed layer were observed in both 1997 and 1998. Round-trip transits of the mixed layer took two-days, timescales long enough to suggest that rotation may be important to the dynamics of convection. When the surface cooling ceased, the water column rapidly restratified, indicating the importance of horizontal processes.

In the Labrador Sea, where cold, fresh water overlies warm, salty water the convecting layer can be visualized as a vertical conveyor belt of heat carrying heat from the warm waters below to the cool atmosphere above without requiring a net change in the convecting layer's temperature. Salinity plays a key role in this transfer. Without the incorporation of additional salt from below, the layer would only deepen with a net temperature loss. Simple heat flux calculations confirm that the majority of the heat content change occurred in water entrained into the convecting layer from below, rather than through a change in the heat content of the convecting layer itself.

Average horizontal velocities in 1997 were almost double those recorded in 1998; however, floats in both years seem to have remained in cold patches of water, water perhaps preconditioned for deep convection, thus biasing the floats toward sampling convective activity. The additional eddy activity noted in 1998 may have been part of a larger time- and space scale phenomenon.

Comparison between ocean heat content changes and NCEP heat fluxes finds good agreement in 1998, but finds NCEP values to be at least 100 W m⁻² low during 1997. NCEP buoyancy flux can predict rms vertical velocity with 83% skill. A linear fit using the nonrotational scaling of (B_oH)^1/3 was found marginally statistically superior to a linear fit with the rotational scaling, (B_o/f)^1/2. Inclusion of both terms did not increase the skill of the model, and the coefficient of the rotational term was indistinguishable from zero.

We conclude that the observations are consistent with vertical nonrotating convection driven by a surface buoyancy flux with a slight influence of other, probably horizontal, effects.