The Dynamics of the Oceanic Air-Sea Boundary Layer

John A. T. Bye ^*

School of Geography, Earth and Atmospheric Sciences, The University of Melbourne, Victoria 3010, Australia

* Correspondence: John A. T. Bye

Academic Editor: Shih-Ang (S. A.) Hsu

Special Issue: Air-Sea Interaction and Marine Meteorology

Received: November 13, 2022 | Accepted: January 25, 2023 | Published: February 05, 2023

Adv Environ Eng Res 2023, Volume 4, Issue 1, doi:10.21926/aeer.2301018

Recommended citation: Bye JAT. The Dynamics of the Oceanic Air-Sea Boundary Layer. Adv Environ Eng Res 2023; 4(1): 018; doi:10.21926/aeer.2301018.

© 2023 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

The importance of wind–wave research from an Australian perspective has been recently highlighted in Greenslade [2] where several aspects of this important topic were discussed, however a background theoretical study of air-sea research in the wave boundary layer was not included in this work.

Here, a background paper is given in which theoretical results on air-sea interaction, based on a similarity model, are presented (Section 2) and compared with various observational studies, both recent and historical (Section 3) with which it is suggested that there is general agreement.

The main feature of this model of the oceanic air-sea boundary layer is that it treats both fluids which occur in the wave boundary layer as a coupled system. In Section 2.1, the similarity relations for the velocity profiles in the air and water are presented, and it is pointed out that in the air the profile arises from aerodynamically rough flow, whereas in the water the profile arises from irrotational wave motions. Section 2.2 makes a clear distinction between the Lagrangian and Eulerian velocity in the water, and Section 2.3 introduces the Ekman and Hasselmann regimes which control the energy transfers from and to the wave field, through two parameters (R and F) which arise respectively from the duration and fetch of the meteorological forcing. Section 2.4 provides an expression for the Charnock parameter in terms of R and F. The surface Stokes velocity is derived in Section 2.5, and the relation between the surface current and the surface Stokes velocity is considered in Section 2.6, which indicates how cancellation of the surface Stokes velocity occurs. Finally, the surface drift velocity in terms of R and F is obtained in Section 2.7. The model thus provides connected theoretical predictions for all the important properties of the air-sea, wind-wave boundary layer.

Section 3 summarizes the observational data, including the classical wave formulae (Section 3.1), wind-wave data (Sections 3.2 and 3.3), and the results from ROMS and SWAN regional modelling (Section 3.4) which are all compared with the predictions of Section 2.

Section 4 concludes that the substitution of F = 1 in the similarity relations, which was used in the original inertial coupling model, is appropriate for large scale ocean modelling studies, and in Section 4.1 the application of the F = 1 model in two reference frames is considered. Section 5 is a brief Conclusion.

2. The Air-Sea Boundary Layer Model

2.1 The Similarity Profile Relations for Air and Water

The longitudinal components of the similarity relations for air and water, presented in Bye and Wolff [3] are,

\[ \operatorname{u}_{\mathrm{Ro}}(\mathrm{z})-\mathrm{u}_\mathrm{o}=\operatorname{Fu}_\mathrm{L}+\mathrm{u}_\ast/\kappa\ln(\mathrm{z}/\mathrm{z}_\mathrm{R}),\mathrm{z}_\mathrm{R}\leq\mathrm{z}\leq\mathrm{z}_\mathrm{B} \tag{1a} \]

\[ \operatorname{u}_{\mathrm{St}}(-\mathrm{z})-\mathrm{u}_\mathrm{o}=\operatorname{\varepsilon u}_\mathrm{L}-\mathrm{w}_\ast/\kappa\ln(\mathrm{z}/\mathrm{z}_\mathrm{R}),\mathrm{z}_\mathrm{R}\leq\mathrm{z}\leq\mathrm{z}_\mathrm{B} \tag{1b} \]

in which it is assumed that the surface shear stress lies along ox (τ_s = τ_s, 0), τ_s > 0 where τ_s = ρ₁u_*² = ρ₂w_*² in which u_* and w_* are respectively the frictional velocities in air and water, and ρ₁ and ρ₂ are respectively the densities of air and water, and oz is vertically upwards in which z = 0 is the mean air-water interface. u_Ro(z) denotes aerodynamically rough flow in the air, and u_St(-z) denotes the Stokes velocity in the water, which has been expressed in terms of a wall boundary layer [4] derived from the truncated Toba spectrum [5].

\[ \phi(\sigma)=\beta \text{u}_* \text{g}\sigma^{-4},\sigma_0\leq\sigma\leq\sigma_1 \tag{2} \]

in which g is the acceleration of gravity and β is a constant, and the frequency, σ = (gk)^1/2 where k is the wavenumber, and in (1a) and (1b), z_R = 1/(2k₁) and z_B = 1/(2k_o) are respectively the high and low wavenumber cut-offs of the spectrum, and κ is von Karman’s constant, ε = (ρ₁/ρ₂)^1/2, and u_o is a reference velocity, which is zero in the Earth’s reference frame. εu_L = u_St(-z_R) - u_o is the wave induced velocity in the water (the surface Stokes velocity) and Fu_L = u_Ro(z_R) - u_o is the wave induced velocity in the air (the spectrally weighted phase velocity). Hence, the ratio of the wave induced velocity in the air to that in the water is F/ε.

At the edge of the wave boundary layer (z = ±z_B), u_Ro(z_B) = u₁ and u_St(-z_B) = u₂ in which u₁ and u₂ are respectively the surface wind and the surface current which are determined by thermohaline processes outside of the wave boundary layer (z_B ≥ z ≥ -z_B). For the Toba spectrum (2), since u_St(-z_B) - u_o = 0,

\[ \varepsilon \text{u}_\mathrm{L}=\varepsilon \text{u}_\mathrm{L})_{\mathrm{To}} \tag{3} \]

where εu_L)_To = w_*/κln(z_B/z_R), >0.

On eliminating u_L between 1(a) and 1(b), the relation,

\[ (\text{u}_1-\text{u}_\text{o})-\text{F}/\varepsilon(\text{u}_2-\text{u}_\text{o})=(1+\text{F})\text{u}_*/\kappa\ln\text{z}_\text{B}/\text{z}_\text{R} \tag{4} \]

is obtained, which on rearranging yields the drag relation,

\[ \text{u}_*=\text{K}_\text{F}{}^{1/2}\bigl(\text{u}_1-\text{u}_\text{o}-\text{F}/\varepsilon(\text{u}_2-\text{u}_\text{o})\bigr) \tag{5} \]

where K_F = ((1 + F)/κlnz_B/z_R)^-2, and the ratio of the surface wind to the surface current (F/ε) is the same as that for the wave induced velocities in the two fluids, and for F = 1, K_F reduces to the inertial drag coefficient,

\[ \mathrm K_\mathrm I=(2/\kappa\ln \ \mathrm z_\mathrm B/\mathrm z_\mathrm R)^{-2} \tag{6} \]

[1,6].

On substituting (6) in (3), the wave induced velocity due to the truncated Toba spectrum,

\[ \varepsilon\mathrm{u}_\mathrm{L})_{\mathrm{To}}=1/2\mathrm{w}_*/\mathrm{K}_\mathrm{I}{}^{1/2} \tag{7} \]

or, alternatively, on eliminating the logarithmic term between (1a) and (1b), yields the relation,

\[ \varepsilon \text{u}_\text{L}=[\varepsilon( \text{u}_1- \text{u}_\text{o})+( \text{u}_2- \text{u}_\text{o})]/(1+ \text{F}) \tag{8} \]

2.2 The Lagrangian and Eulerian Shears in the Water

On substituting for (u₂ - u_o) in (8) from the relation,

\[ \varepsilon \text{u}_\text{L}+ \text{r}( \text{u}_2- \text{u}_\text{o})=0 \tag{9} \]

which was introduced in Bye and Wolff [6] to relate the Lagrangian shear (εu_L) and the Eulerian shear (u₂ - u_o) in the oceanic boundary layer, we obtain, u₁ - u_o = u_L(1 + F + 1/r), and from (1a) evaluated at z = z_B, using (7), u₁ - u_o = Fu_L + ½u_*/K_I^1/2, from which, on eliminating u_L, it is found that,

\[ \text{u}_*=\text{K}_\text{I}{}^{1/2}(\text{u}_1-\text{u}_\text{o})/\text{R} \tag{10} \]

where

\[ \text{R}=1/2\bigl(1+\operatorname{Fr}/(1+\operatorname{r})\bigr) \tag{11a} \]

and

\[ \text{r}=(2\text{R}-1)/(1+\text{F}-2\text{R}) \tag{11b} \]

Alternatively on substituting for (u₁ - u_o) in (10) yields,

\[ \varepsilon \text{u}_\text{L}/\text{w}_*=1/2(2\text{R}-1)/\left(\mathrm{FK_I}^{1/2}\right) \tag{12} \]

Eq. (10) enables R to be evaluated in terms of observable quantities. Thus, on making use of the logarithmic velocity profile in the air (10), may be expanded to yield, (u₁₀ - u_o) + (u₁ - u₁₀) = Ru_*/K_I^1/2, which yields the relation,

\[ \mathrm{R}=\mathrm{K_I}^{1/2}\left(\mathrm{K_{10}}^{-1/2}+\kappa^{-1}\ln(\mathrm{z_B}/10)\right) \tag{13} \]

where z_B = gT²/8π² in which T = 2π/(gk_o)^1/2 is the peak wave period. Eq. (13) is identical with the expression for R for F = 1 in Bye and Wolff [7], and is shown in Figure 1. The important feature of Figure 1 is that R increases with T at a constant u_*. This occurs as the duration of the wind forcing increases and swell is developed.

Figure 1 The parameter (R) as a function of u_* and T evaluated from equation (A1) in the Appendix of Bye et al. [8] with K_10m = 0.002 and u_10m = 40 ms^-1 and reproduced from Figure 2 of Bye et al. [7].

On substituting for εu_L from (12) in (9) using (11b), It is also found that,

\[ \mathrm{u}_2-\mathrm{u}_\mathrm{o}=\text{w}_*\big(\mathrm{R}-1/2(1+\mathrm{F})\big)/\big(\mathrm{FK}_\mathrm{I}{}^{1/2}\big) \tag{14} \]

2.3 Ekman and Hasselmann Regimes

Eq. (14) is of high significance as for R > R_c where,

\[ \mathrm{R}_\text{c}=1/2(1+\mathrm{F}) \tag{15} \]

u₂ - u_o has the same sign as w_*, and hence the surface current can support an Ekman regime, whereas for R < R_c u₂ - u_o opposes the surface shear stress and an Ekman regime is not supported and instead a Hasselmann regime occurs. We use these names to emphasize the physical difference between the two regimes. In the Ekman regime there is a net loss of energy from the wave field to the deep ocean, whereas in the Hasselmann regime there is a net gain of energy from the atmosphere to the wave field. There is no net transfer of energy to or from the wave field for R = R_c, which is the condition for the fully developed sea. For F = 1, R_c = 1.

On now turning to the evaluation of the wave induced velocity in the air, the spectrally weighted phase velocity of the gravity waves due to the truncated Toba spectrum is,

\[ \operatorname{Fu}_\mathrm{L})_{\mathrm{To}}=\mathrm{g}/\sigma_\mathrm{M} \tag{16} \]

where σ_M² = ∫^σ₁,^σ_oσ²φ(σ)dσ/∫^σ₁,^σ_oφ(σ)dσ, from which,

\[ \mathrm{Fu}_\mathrm{L})_{\mathrm{To}}=\mathrm{c}_\mathrm{o}\big[\big(1+\mathrm{k}_\mathrm{o}/\mathrm{k}_1+(\mathrm{k}_\mathrm{o}/\mathrm{k}_1)^{1/2}\big)/3\big]^{1/2} \tag{17} \]

in which c_o is the phase speed of the peak wave [3], and since c_o > 0, F > 0. On expressing the cut-off wavenumber ratio (k_o/k₁) in terms of the inertial drag coefficient (K_I) from (6), k_o/k₁ = exp(-½κK_I^-1/2), and hence on substituting for k_o/k₁ in (17) with κ = 0.40 and K_I = 0.0015 [6], it is found that,

\[ \operatorname{Fu}_\mathrm{L})_{\mathrm{To}}=0.60\mathrm{c}_\mathrm{o} \tag{18} \]

which, on substituting for u_L)_To from (7), yields the expression for the wave age,

\[ \mathrm{c}_\mathrm{o}/\mathrm{u}_*=21.5\mathrm{F} \tag{19} \]

where 1/(1.20K_I^1/2) = 21.5.

2.4 The Charnock Parameter

Eq. (19) can be used to obtain a prediction for the Charnock parameter (α), defined by the relation, z_o = αu*²/g, in which z_o is defined relative to the reference velocity, u_o [9]. Using the logarithmic identity lnz_o/z_B = lnz_R/z_B + lnz_o/z_R, in which from (6), the first term, lnz_R/z_B = -½κ/K_I^1/2, and from (1a) evaluated at u = u_o, on substituting for εu_L from (12), the second term, lnz_o/z_R = -½(2R - 1)κ/K_I^1/2. Hence, z_o/z_B = exp(-κRK_I^-1/2), which on expressing z_B in terms of the wave speed of the peak wave (c_o) yields,

\[ \alpha=1/2(\mathrm c_o/ \text{u}_\ast)^2\exp\left(-\kappa\mathrm R\mathrm {K_I}^{-1/2}\right) \tag{20} \]

Eq. (20) is an expression for the Charnock parameter [9] which for R = 1 depends only on the inertial drag coefficient and the wave age [6], or alternatively, on substituting (19) in (20),

\[ \alpha=1/2(21.5\mathrm{F})^2\exp\left(-\kappa\mathrm{RK_I}^{-1/2}\right) \tag{21} \]

Eq. (19) is a direct prediction for c_o/u_* in terms of F, and (21) is a prediction for α in terms of the two parameters (F and R) which arise respectively from the fetch variability and the duration variability of the wave field.

2.5 The Surface Stokes Velocity

On evaluating the ratio of the wave induced velocity due to the truncated Toba spectrum (εu_L)_To from (7) to the wave induced velocity (εu_L) from (12), the ratio,

\[ \varepsilon \text{u}_\mathrm{L})_\mathrm{To}/\varepsilon \text{u}_\mathrm{L}=\mathrm{F}/(2\mathrm{R}-1) \tag{22} \]

and hence, on substituting for F from (15), it is found that,

\[ \varepsilon \text{u}_\text{L})_{\text{To}}/\varepsilon \text{u}_\text{L}=(2 {\mathrm{R}}_\text{c}-1)/(2 {\mathrm{R}}-1) \tag{23} \]

which indicates that the Toba truncated spectral model, εu_L)_To, underestimates the wave induced velocity (εu_L) in the Ekman regime and overestimates it in the Hasselmann regime, It is suggested that the physical processes which gives rise to this response are that in the Ekman regime, the transfer of energy from the wave field to the deep ocean occurs through the promotion of long period waves which, as pointed out in Bye and James [10], would lead to a low wavenumber tail, which is absent in the truncated Toba spectral model. Conversely, in the Hasselmann regime, the transfer of energy from the atmosphere to the wave field is promoted by long period waves which are over estimated by the truncated Toba spectral model.

2.6 The Relation between the Surface Stokes Velocity and the Surface Current

The sum of the surface Stokes velocity and the surface current, relative to the reference velocity, εu_L + (u₂ - u_o), can be expressed solely in terms of εu_L using (9) and (11b) which yields,

\[ \varepsilon \text{u}_\text{L}+(\text{u}_2-\text{u}_\text{o})=(4\mathrm{R}-2-\mathrm{F})/(2\mathrm{R}-1)\varepsilon \text{u}_\text{L} \tag{24} \]

Hence the sum of the wave induced and turbulent velocities (εu_L + u₂ - u_o) is zero for R = R_can, where,

\[ \mathrm{R}_{\mathrm{can}}=1/4(\mathrm{F}+2) \tag{25a} \]

such that for F = 1, the surface current cancels the surface Stokes velocity when R = ¾ and also the ratio,

\[ \mathrm{R}_{\mathrm{can}}/\mathrm{R}_{\mathrm{c}}=(1+1/2\mathrm{F})/(1+\mathrm{F}) \tag{25b} \]

Hence from (25b) the cancellation always occurs in the Hasselmann regime, and at low fetches (F → 0), since R_can/R_c → 1, it can occur with no net energy transfer to or from the wave field as postulated in [11].

More generally, from (24), the ratio of the Eulerian and Lagrangian shears,

\[ (\text{u}_2-\text{u}_\text{o})/\varepsilon \text{u}_\text{L}=1-(2\mathrm{R}_\text{c}-1)/(2\mathrm{R}-1) \tag{26a} \]

\[ =1-\operatorname{F}/(2\operatorname{R}-1) \tag{26b} \]

which is a measure of the titanic struggle of Euler and Lagrange in determining the current outcome, that is occurring as F and R vary. The ratio of the Eulerian and Lagrangian shears is negative in the Hasselmann regime, positive in the Ekman regime and zero in the fully developed sea.

The increase in the relative importance of (u₂ - u_o) in the Ekman regime as R increases is of particular significance as this wave-current interaction increases the transfer of energy from the wave field to the deep ocean. This important process does not appear to have been highlighted previously in advanced wave modelling studies, e.g. Babanin et al [12].

2.7 The Surface Drift Velocity

The matching of the air and water velocities occurs through the drift velocity.

On assuming that the velocities in the two fluids are continuous across the interface, and equating the velocities in 1(a) and 1(b), we obtain, u_*/κ lnz_s/z_R = (ε - F)u_L/(1 + ε) in which z_s is the roughness length at the edge of the drift layer, from which,

\[ \text{u}_\text{s}-\text{u}_\text{o}=(1+\text{F})\varepsilon\text{u}_\text{L}/(1+\varepsilon) \tag{27} \]

where (u_s - u_o) is the drift velocity, which on using (12) may also be expressed as,

\[ \text{u}_\text{s}-\text{u}_\text{o}=1/2(2\mathrm{R}-1)(1+\text{F})/\left(\text{FK}_\text{i}{}^{1/2}\right)\text{w}_\ast/(1+\varepsilon) \tag{28} \]

For F = 1, from (27), the ratio of the surface drift velocity to the surface Stokes velocity, relative to the reference velocity,

\[ ( \text{u}_ \text{s}- \text{u}_ \text{o})/\varepsilon \text{u}_ \text{L}=2/(1 + \varepsilon) \tag{29} \]

3. Observations

There are several types of observations with which the results of the model can be compared. These observational results are considered below. Special emphasis will be placed on, under which conditions, F = 1, which was proposed in the original inertial coupling analysis [1], is a good approximation.

3.1 Classical Wave Formulae

In the early wave models, the peak wave period (T) was predicted in terms of the 10 m wind speed (u₁₀) and also the fetch and duration. These models lead to a dimensional expression of the form,

\[ \mathrm T=\mathrm A_ \mathrm o \text{u}_{10}/ \mathrm g \tag{30} \]

where g is the acceleration of gravity and A_o is an observationally determined constant, which may depend on fetch and duration, and asymptotes to a limiting value at large fetches and durations. On expressing (30) in terms of the friction velocity in air (u_* = (τ_s/ρ₁)^1/2) where τ_s is the surface shear stress and ρ₁ is the air density, we obtain,

\[ \mathrm{c}_\mathrm{o}/\mathrm{u}_*=\mathrm{a}_\mathrm{o} \tag{31} \]

in which a_o = A_o/(2πK₁₀^1/2), where K₁₀ is the 10 m drag coefficient. Eq. (31) has the form of the wave age (19).

In a coastal sea, observations for the fully developed sea indicate that A_o = 8.6 [13] from which, using a 10 m drag coefficient, K₁₀ = 0.002, we obtain, a_o = 30.6, and hence on substituting for c_o/u_* from (31), at large fetches, F = F_max where F_max = 1.42. Thus the similarity model (1a and b) would apply over the range of surface parameter (0 ≤ F ≤ F_max) in which at small fetches F → 0. Fetch dependent observations also indicate that the peak period (T) is proportional to x^1/3 where x is the fetch [14], hence the mean peak period over the fetch length is ¾ T from which it follows that the predicted value for the mean conditions is F is 1.07, which is very similar to the condition, F = 1.

In summary, the model, with F = 1, applies for the mean values of u_* and c_o which occur over the fetch length (x).

3.2 Wind-wave Studies

As was seen in Section 3.1, the variability of F in the coastal sea can be attributed to a fetch dependence of the fully developed wave field. A similar variability would occur in more general wave conditions. This variability has been observed in wind-wave studies, and is conveniently documented in Figure 1 of Bye et al. [15], which is reproduced from Figure 12 of Donelan et al [16] on to which contours of R using (13) have been added. The data points from the various wind-wave studies are shown as a function of inverse wave age (u_*/c_o) and R. For each data point, F can be determined from (19) and hence R_c from (15).

The main group of observational points, which excludes the HEXOS, wave tank and Lake Ontario data groups, has the approximate parameter values of R ≈ 0.95 and c_o/u_* ≈ 20, which yield F ≈ 0.93 and R_c ≈ 0.97.

The parameters for the cluster of points from the HEXOS data set are R ≈ 0.75 and c_o/u_* ≈ 11.8. Hence F ≈ 0.55 and R_c ≈ 0.78, and the Lake Ontario group, has the approximate parameter values of R ≈ 0.6 and c_o/u_* ≈ 5, from which F ≈ 0.23 and R_c ≈ 0.62.

Hence R < R_c for the three data groups, which indicates that in the wind-wave studies, a Hasselmann regime occurred. This was biased towards the low end of wave development at the fetches prevailing in all three data groups in which F < 1, in particular for the HEXOS and Lake Ontario data sets (F << 1).

3.3 The Cancellation of the Surface Drift by the Surface Current

In the Lake Ontario group of the wind-wave observations in Section 3.2, F ≈ 0.23, and R ≈ 0.6, which is similar to the theoretical prediction for cancellation (25a) of R_can ≈ 0.56, i.e. close to the cancellation condition, and is also similar to R_c ≈ 0.62. Thus here we have a regime in which there is almost no net energy transfer to or from the wave field and also cancellation of the surface drift by the surface current is occurring, as shown in Section 2.6 for F → 0 This is an exceedingly interesting and unusual observational situation, which closely mirrors the prediction for cancellation proposed by Hasselmann [11] which is discussed in Section 2.6.

3.4 Wave Transport in the South Australian Basin

In a recent wind-wave study [10] for the South Australian Basin during the period 2010-2012 the wave environment was dominated by swell, generated in the storm systems, with a monthly average peak wave period (T) of about 11 s (c_o = 17 ms^-1, z_B = 15 m), and the median values obtained from histograms of R over daily averages from the Regional Ocean Modelling System (ROMS) using (15) was R = 1.19, and from the Simulating Waves Nearshore (SWAN) using (16) with F = 1 to evaluate u₂ and v₂ = -u_{2 ,} was R = 1.17. The similar magnitudes of these two independent estimates of R strongly supports the existence of an Ekman regime in which F = 1.

The daily (i) SWAN spectral transport U_sw(i) = ∫^∞,oσφ_i(σ)dσ) and the Toba spectral transport (U_To(i) = z_Bw_*/κ)_i) were also evaluated over the two year period, 2011–2012 [10]. Figure 2 shows that at low values there is a cluster of data points in which U_sw(i) and U_To(i) are approximately equal and hence from (23), since F = 1, R = 1 as found for the mean wave conditions in the classical wave studies (Section 3.1), At higher values, however, there is a pronounced linear trend in which U_sw(i) = 1.36U_To(i) as predicted by (22) for the observed median parameter, R = 1.18 and F = 1. The regression slope of best fit (1.16) is a split between the data points of the two physical models. It is concluded, on the assumption that the transports are proportional to the wave induced velocity in the water (εu_L), that the cluster of low value points in Figure 2 is due to the fully developed coastal circulation in which R = 1 (Section 3.1) and more importantly that the trend in the higher value points is consistent with open ocean conditions in which the predicted ratio (23) is applicable.

Figure 2 The linear regression of the daily SWAN spectral transport and the daily truncated Toba spectral transport for the period 2011-2012 in the South Australian Basin reproduced from Figure 9 of Bye and James [10], with the inclusion of the linear relation (22).

3.5 The Fully Developed Sea

The concept of the fully developed sea is well known, see for example, Jones and Toba [17]. In the model, from (15), the fully developed sea occurs for R = ½(1 + F), from which the wave age (c_o/u_*) and the Charnock parameter (α) in the limit of large fetches can be evaluated by substituting F_max = 1.42 in (19) and (21) respectively. The results are, from (19), c_o/u_*)_max = 30.5, which is similar to the observational estimate of 34 [5], and from (21) α_o)_max = 0.017 which is similar to the observational estimate of 0.019 [18].

4. The Inertial Coupling Relation

The observations reported in Sections 3.1 and 3.4, and in the main group of points in Section 3.2 are consistent with F ≈ 1, as was proposed in the original paper on inertial coupling [1]. The observations are representative of oceanic and coastal environments. The data in Section 3.2 suggest that in more confined environments, F << 1, as indicated in Section 3.1.

The underlying physics which leads to these results is:

The two-dimensional turbulence of the atmosphere imposes a uniform spatial scale on the ocean, that can be interpreted as a ‘moving’ fetch, and hence a uniform, F, in which the kinetic energy is partitioned at the sea surface, such that in the absence of a surface shearing stress (τ_s) the kinetic energies of the atmospheric wind and the oceanic current would be expected to be equal. Thus, for τ_s = 0, ρ₁(u₁ - u_o)² = ρ₂(u₂ - u_o)², which is satisfied by (5) with F = 1, which reduces to the inertial coupling relation,

\[ \text{u}_*= \text{K}_I{}^{1/2}\left(\rho_1{}^{1/2}( \text{u}_1- \text{u}_ {o})-\rho_2{}^{1/2}( \text{u}_2- \text{u}_ {o})\right)/\rho_1{}^{1/2} \tag{32} \]

On this basis we propose that the inertial coupling relation (32), which follows from the substitution of F = 1 in the similarity relations in the wave boundary layer, is of general applicability in large scale theoretical oceanic studies.

4.1 Reference Frames

The significance of two reference velocity (u_o) will now be considered.

4.1.1 The Surface Geostrophic Reference Frame

This is a very useful reference frame in which u_o = u_go where u_go is the surface geostrophic velocity, which separates the frictional components from the geostrophic components. Here (32) may be expressed in the form,

\[ \text{u}_*= \text{K}_\mathrm{I}{}^{1/2}( \text{u}_{\mathrm{e}1}- \text{u}_{\mathrm{e}2}/\varepsilon) \tag{33} \]

where u_e1 = u₁ – u_go and u_e2 = u₂ – u_go. The vector version of this reference frame (u_o = u_go), taking account of the horizontal variability of u_go, has been used by the author in an ongoing research projects (An integrated theory of the air-sea boundary layer: Part I Surface friction on the large scale ocean circulation. Unpublished ms.).

The surface reference frame (u_o = u₂) could also be used, provided that the complex horizontal variability of u₂ can be taken into account.

4.1.2 The Earth Reference Frame

This fundamental reference frame in which u_o = 0 is of course used implicitly in most studies. The theoretical results of Section 2 use this reference frame although for completeness, u_o, is retained in the derivations.

5. Conclusions

The main conclusion is that it is possible to successfully predict the major properties of the wave boundary layer using a similarity representation for the air and the water velocity profiles. The model predicts the properties in terms of two parameters (F and R) which depend respectively on the fetch and duration of the wave environment, and importantly shows that F = 1, which was assumed in the original inertial coupling relation for the air-sea boundary layer, is a good approximation for large scale ocean modelling studies. Hence, the important parameter is R, which increases in swell dominant environments (Section 3.4) and controls the input of energy from the wave field into the deep ocean. This process can be further examined in global eddy resolving ocean circulation modelling studies.

Author Contributions

The Author is solely responsible for this work.

Competing Interests

The author has declared no competing interests exists.