Momentum and heat transfer across the foam-covered air-sea interface in hurricanes

Abstract

Procedures of formal averaging over the air-sea interface are applied to both momentum and enthalpy surface-transfer coefficients, C_D and C_K, in hurricane conditions. The transfer coefficients across the total area of the sea surface and water-covered portions of the sea surface, C_D, C_K, and C_Dw, C_Kw, are estimated by measurements in the open-sea and laboratory (foam-free) conditions, respectively, while the transfer coefficients across the foam-covered portion of the sea surface, C_{Df, Kf}, which cannot be measured directly, are estimated using the splitting relations. Applying the Monin-Obukhov similarity theory at the neutral stability atmospheric conditions to the transfer coefficients (separately for the foam-free, foam-covered, and total sea surfaces) yields the roughness lengths Z_Dw, Z_Kw, Z_Df, Z_Kf, and Z_D, Z_K. The study is aimed at the description of an anomalous, as compared with laboratory conditions, behavior of the momentum and enthalpy transfer coefficients in hurricane conditions with wind speed U₁₀ by the effect of the foam slipping layer sandwiched between the atmosphere and the ocean.

Appendix A. Averaging model of the heat transfer across the air-sea interface

The turbulent fluxes of momentum, τ, sensible heat, Q_H, and latent heat, Q_E, in the surface layer between the atmosphere and ocean are defined by bulk formulas (e.g., Garratt (1992); Andreas et al. (2012); Zou et al. (2017); Komori et al. (2018), and references therein):

$$ \tau =\rho {C}_D{U}_{10}^2, $$

(15)

$$ {Q}_H=\rho {c}_p{C}_H{U}_{10}\Delta {\theta}_{10}, $$

(16)

$$ {Q}_E=\rho {L}_V{C}_E{U}_{10}\Delta {q}_{10}, $$

(17)

where ρ is the air density; C_D, C_H, and C_E are the transfer coefficients of the momentum, sensible heat, and latent heat, respectively; c_p is the specific heat of humid air at a constant pressure; L_V is the latent heat of vaporization; ∆θ₁₀ = θ₀ − θ₁₀, and ∆q₁₀ = q₀ − q₁₀; U₁₀, θ₁₀, and q₁₀ are the values of wind speed, temperature, and specific humidity at the 10 [m] reference height, respectively; and θ₀ and q₀ are the temperature and the specific humidity at the sea level. Then

$$ {\overline{Q}}_N=\frac{1}{S}\underset{S}{\int }{Q}_N dS^{\prime}\equiv \frac{1}{S}\left(\underset{S_w}{\int }{Q}_{Nw}d{S}^{\prime }+\underset{S_f}{\int }{Q}_{Nf}d{S}^{\prime}\right), $$

(18)

where N = H, E; S′ is the variable of integration along the sea surface; the overbars indicate the values averaged over a sea surface; S_w, S_f, and S = S_w + S_f are the areas of the foam-free, foam-covered, and total sea surfaces, respectively. It should be noted that this averaging procedure was used in the analyses of microwave emissivity (Stogryn 1972) or temperature (Guimbard et al. 2012) of the foam-covered sea surface.

Then, the transfer coefficients ${\overline{C}}_N$={${\overline{C}}_H$ or ${\overline{C}}_E$} in Eqs. (15–17) can be split into the sums of their partial values over the foam-free and foam-covered sea surfaces weighted with the foam coverage coefficient, α_f:

$$ {\overline{C}}_N=\left(1-{\alpha}_f\ \right){\overline{C}}_{Nw}+{\alpha}_f{\overline{C}}_{Nf}. $$

(19)

Here

$$ {\overline{Q}}_{Hw}=\rho {c}_p{\overline{C}}_{Hw}{U}_{10}\Delta {\theta}_{10},\kern0.75em {\overline{Q}}_{Hf}=\rho {c}_p{\overline{C}}_{Hf}{U}_{10}\Delta {\theta}_{10}{\overline{,Q}}_H=\rho {c}_p{\overline{C}}_H{U}_{10}\Delta {\theta}_{10}, $$

(20)

$$ {\overline{Q}}_{Ew}=\rho {L}_V{\overline{C}}_{Ew}{U}_{10}\Delta {q}_{10},\kern1em {\overline{Q}}_{Ef}=\rho {L}_V{\overline{C}}_{Ef}{U}_{10}\Delta {q}_{10},\kern0.75em {\overline{Q}}_E=\rho {L}_V{\overline{C}}_E{U}_{10}\Delta {q}_{10} $$

(21)

are the corresponding averaged values of heat and moisture fluxes across the foam-free, foam-covered, and total sea surfaces, respectively. As commonly accepted now, the latent and sensible heat transfer coefficients have the same shape as the enthalpy coefficient as a function of wind speed U₁₀ (Zhang et al. 2008; Bao et al. 2011; Komori et al. 2018)

$$ {\overline{C}}_K\left({U}_{10}\right)={\overline{C}}_H\left({U}_{10}\right)={\overline{C}}_E\left({U}_{10}\right). $$

Then, the latent and sensible heat transfer coefficients can be represented by the enthalpy coefficient, and we can set N = K. Furthermore, α_f = S_f/S is the foam coverage approximated as

$$ {\alpha}_f=\gamma \tanh \left[\alpha\ \exp \left(\beta \frac{U_{10}}{\ {U}_{10}^{(S)}}\right)\right] $$

(22)

with α = 0.00255, β = 8, γ = 0.98, and ${U}_{10}^{(S)}\approx 48\ \left[m{s}^{-1}\right]$. The above formula proposed by Holthuijsen et al. (2012) approximates observation data for α_f up to U₁₀ < 50 [ms⁻¹] and expands for U₁₀ > 50 [ms⁻¹] by choosing the above values, β, γ, and ${U}_{10}^{(S)}$using a least-squares approximation in GS2016; GS2018.

The logarithmic velocity profile is commonly accepted for the neutral atmosphere for any underlying surfaces. Therefore, it is assumed in the present study that the velocity profiles over the foam-free, foam-covered portions of the sea surface, and the total sea surface, respectively, are logarithmic ones. It is also assumed that the transfer coefficients ${\overline{C}}_D$ and ${\overline{C}}_K$ across the total sea surface are estimated in the open-sea conditions. Whereas the transfer coefficients across the foam-free portions of the sea surfaces can be directly estimated in laboratory conditions as ${\overline{C}}_K={\overline{C}}_{Kw}$, the foam input in ${\overline{C}}_K$ is negligibly small. Since the transfer coefficients over the sea surface portion completely covered with foam ${\overline{C}}_{Kf}$ cannot be measured directly in the whole region of wind speed variations, the splitting Eq. (19) can be used for their definition.

According to the Monin-Obukhov similarity theory, the bulk transfer coefficients are related to atmospheric stability and reference height of measurements. To eliminate the influence of these factors, the bulk transfer coefficients are reduced to neutral stability atmospheric conditions with a standard 10 [m] reference height (e.g., Andreas et al. (2012); Zou et al. (2017); Sergeev et al. (2017), and references therein). As it is noted in Hsu (2003) and Hsu et al. (2017), the logarithmic vertical profile of the mean wind speed can be adopted in real hurricane conditions under near-neutral stability atmospheric conditions. This results separately to the foam-free, foam-covered, and total sea surfaces

$$ {\overline{C}}_{Dw}={\left(\frac{\kappa }{\mathit{\ln}\left[{Z}_{10}/{Z}_{Dw}\right]}\right)}^2,\kern1em {\overline{C}}_{Df}={\left(\frac{\kappa }{\mathit{\ln}\left[{Z}_{10}/{Z}_{Df}\right]}\right)}^2\ {\overline{C}}_D={\left(\frac{\kappa }{\mathit{\ln}\left[{Z}_{10}/{Z}_D\right]}\right)}^2, $$

(23)

$$ \frac{{\overline{C}}_{Kw}}{\sqrt{{\mathrm{C}}_{Dw}}}=\frac{\kappa }{\mathit{\ln}\left[{Z}_{10}/{Z}_{Kw}\right]},\kern0.5em \frac{{\overline{C}}_{Kf}}{\sqrt{{\mathrm{C}}_{Df}}}=\frac{\kappa }{\mathit{\ln}\left[{Z}_{10}/{Z}_{Kf}\right]},\frac{{\overline{C}}_K}{\sqrt{{\mathrm{C}}_D}}=\frac{\kappa }{\mathit{\ln}\left[{Z}_{10}/{Z}_K\right]}\cdotp $$

(24)

Obviously, Eqs. 23 and 24 at known values of the transfer coefficient in their left-hand sides can be regarded as definitions of the roughness length parameters (Z_K, Z_Kw, and Z_Kf) across the corresponding portions of the sea surfaces (the bars over the corresponding variables are omitted throughout the remaining text without confusion).