ON THE COOL SKIN OF THE OCEAN

JIN WU

Air-Sea Interaction Laboratory, College of Marine Studies, University of Delaware. Lewes, Delaware 19958. U.S.A.

(Received in final form 24 August, 1984)

Abstract. Previous data relating sea-surface temperature to heat flux across the air-sea interface were reanalyzed with a common formula for the wind-stress coefficient. An expression is proposed for the nondimensional thickness of the thermal sublayer: the expression increases with wind velocity at light winds and has a value of 7 when the wind velocity reaches 7 m s - ‘.

1. Introduction

A thin region of viscous flow exists immediately below the sea surface; near the upper boundary of this region is a thermal sublayer. As the sea surface is generally cooler than the subsurface water, the thermal sublayer within which the temperature change is concentrated is commonly called the cool skin (Roll, 1965). The study of the thermal sublayer is important on the one hand in predicting heat exchange between the atmosphere and the ocean, and on the other in determining the sea-surface temperature, which serves as a predictor for long-range weather patterns.

Since the heat transfer near the air-sea interface is primarily through a molecular process, the temperature of, and the heat flux across, the interface are related to the thermal-sublayer thickness (Saunders, 1967). Its nondimensional thickness was sug- gested on the basis of studies over solid surfaces to have a constant value (Wu, 1971). This suggestion, however, is not supported by recent oceanic data (Grassl, 1976; Simpson and Paulson, 1980; Paulson and Simpson, 1981), especially when the data are reanalyzed with a common formula for the wind-stress coefficient. An expression is proposed for the nondimensional thickness of the thermal sublayer on the basis of the corrected data.

2. Fundamentals

The flow in the upper layer (above the seasonal thermocline) of the ocean has been found to be turbulent almost all the time (Grant et al., 1968). The aqueous boundary layer is turbulent with a viscous sublayer immediately below, and undulating with, the sea surface. Within the sublayer, exchanges of momentum and heat are mainly due to molecular processes (Ewing and McAlister, 1960; McAlister and McLeish, 1969; Khundzhua et al., 1977). Relative efficiency of momentum and heat transfers depends on the molecular Prandtl number (Schlichting, 1968): P = v/a, where v and a are respectively the kinematic viscosity and molecular thermal diffusivity of the fluid. For water with P = 7 approximately, the momentum diffusion is therefore more effective

Boundary Layer Meteorology 31 (1985) 203-207. OOOS-8314/85/0312-0203.SOO.75. 0 1985 by D. Reidel Publishing Company.

204 JIN WU

than the thermal diffusion. Consequently, we have inside the viscous sublayer a region where the heat transfer is through the molecular process; this region is the so-called thermal sublayer. For water flowing over a heated flat plate at zero incidence, the thickness of the thermal sublayer is approximately one half that of the viscous sublayer.

The thickness of thermal sublayer (S,) can be related to net heat flux (Q) across the air-sea interface (through the sublayer) as (Saunders, 1967),

6 = hAT I

Q (1)

where AT is the difference in temperatures at the sea surface and at the lower boundary of the sublayer; cP and p are respectively the specific heat and density of water. Following studies over a solid surface, the thickness of the sublayer below the sea surface was considered to be inversely proportional to the shear stress (r) applied by currents on the underside of the interface,

&=a$ 5 112

u*= - U* 0 P

where a is a proportionality constant, and U* the friction velocity of currents. Over a solid boundary, the nominal thickness of the viscous sublayer was found (Schlichting, 1968) to be S, = 11.6v/u*. Consequently, the proportionality constant a was suggested to have a value of 5.8 (Wu, 1971).

The friction velocity of aqueous flows is generally unknown but can be estimated from the wind stress. The shear stress acting on the underside of the interface is, however, smaller than the wind stress on the upper side (Stewart, 1961). In order to accommodate this discrepancy, a coefficient I was incorporated by Saunders (1967) as

Q v A. Qv &‘-=u-= ~ cp PM* c,pu;

J=g”;;, u*

u; = u**(%)‘~2, U*, = (y’ (3)

where u*, is the friction velocity of the wind, r, the wind stress, p, the density of air, and ~4 the current-friction velocity calculted from z = ra. Subsequently, field experi- ments have been conducted to determine the value of A.

3. Reanalysis of Oceanic Results

3.1. DATA

Oceanic data reported by various investigators are compiled in Figure la, where U,, is the wind velocity measured at the standard anemometer height (10 m) above the mean

ON THE COOL SKIN OF THE OCEAN 205

(al

CD 0

- 0

0

0

8 00

0 Q

@

00 0 0

a0 .

0

0

0 0

-

Q 0 -

-

b) 0

00 O c!E

0 2 4 6 8 10 12

Wind Velocity, U,. (m s ’ )

Fig. 1. Frequently cited oceanic results on coefficients of thermal-sublayer thickness. The data are from Grass1 (1976), (0); Simpson and Paulson (1980), (0); and Paulson and Simpson (1981),(a). The original

data are shown in (a), and the values corrected with a common expression of C,, in (b).

206 JIN WL

sea surface. These are the available sets of data, which cover a wide range of wind velocities.

Grass1 (1976) measured the temperature difference between the sea surface and j-cm depth, and determined two values of 2 for each wind velocity by adopting two different expressions for the wind-stress coefficient; his results obtained with the wind-stress coefficient of 1.3 x 10 - 3 are presented in the figure. The results of Simpson and Paulson (1980) were already presented in terms of U,,. Their most recent data (Paulson and Simpson, 1981) presented in terms of U& were recovered through calculations with pdc2 = pal&, and the wind-stress coefficient of 1.4 x 10e3 used by them.

3.2. TRENDS

Various values of the wind-stress coefficient were adopted by different investigators in their data analysis. For a fair comparison, their results were recalculated by adopting uniformly the wind-stress coefficient suggested by Wu (1980). The recalculated values of 2, shown in Figure la, are presented in Figure lb, and are seen to be much less scattered.

The results shown in Figure 1 have a rather clear trend: 2 increasing with U,, at low winds and approaching a constant value at high winds. The scattered data do not warrant a detailed curve fitting; two expressions in simplest forms are proposed to approximate the trend:

;1 = 2 + (5/7)U,,, U,,<7ms-’

(4) /l=7, 7msp'<U,,<12ms-'

where U,, is expressed in m s - ‘. Lines corresponding to the above expressions are drawn in Figure lb. Note that the proposed expressions are consistent with our experimental results (Wu, 1984) on the variation of viscous-sublayer thickness with wind velocity.

3.3. JUSTIFICATIONS

The existence of the viscous sublayer is due to the damping of turbulence at the boundary. At a solid surface, the fluid motions vanish all together. Accessing primarily due to horizontal motions at the air-sea interface in contrast to nonslip conditions at the solid surface, Saunders (1973) suggested that turbulence should extend closer to the interface than to the solid surface. We suspect that vertical surface undulations also make the flow near the interface less stable than that over a fixed boundary, again implying a closer penetration of turbulence. The thickness of the viscous sublayer, and consequently that of the thermal sublayer, at the interface are therefore thinner. Previous measurements of the viscous sublayer (Wu, 1975; McLeish and Putland, 1975) substantiated the thinning effects of the interfacial characteristics on the sublayer. Our recent results (Wu, 1984) indicate that the thinning is relatively more effective at lower wind velocities where the sublayer is thicker.

In practice, the temperature deviation was obtained from the sea-surface and bulk-water temperatures; in the formulation, the temperature at the lower boundary of

ON THE COOL SKIN OF THE OCEAN 207

the thermal sublayer was used, instead of the bulk-water temperature obtained at a much greater depth. This discrepancy results in a larger temperature difference in measure- ments than in the formula. Consequently, when thinning effects of interfacial character- istics become negligible at high winds, the value of 1 can be greater than the solid-surface value, 5.8.

4. Concluding Remarks

The thermal sublayer of the ocean has been considered to have a constant non- dimensional thickness; most related measurements have been customarily interpreted in this context. The suggested variation of this thickness with wind velocity may alter our analyses of oceanic measurements, especially those on the heat flux from the ocean and the sea-surface temperature deviation.

Acknowledgement

I am very grateful for the sponsorship of this work provided by the Mechanics Division, Office of Naval Research, under Contract N00014-83-K-0316, and the Physical Oceanography Program, National Science Foundation under Grant No. OCE-8214998.

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