Interpretation of coastal wind transfer functions with momentum balances derived from idealized numerical model simulations

Abstract

The local wind-driven circulation off southern San Diego is addressed with two complementary statistical and dynamical frameworks based on observations and idealized numerical model simulations. The observations including surface currents from high-frequency radars, subsurface currents from a nearshore mooring, and wind records at a local wind station are analyzed with the idealized ocean model (MITgcm) simulations using realistic bottom topography and spatially uniform wind stress forcing. Statistically estimated anisotropic local wind transfer functions characterize the observed oceanic spectral response to wind stress separately in the x (east-west) and y (north-south) directions. We delineate the coastal circulation at three primary frequencies [low (σ _L=0.0767 cycles per day (cpd)), diurnal (σ _D=1 cpd), and inertial (σ _f=1.07 cpd) frequencies] with the momentum budget equations and transfer functions. At low frequency, the magnitudes of transfer functions are enhanced near the coast, attributed to geostrophic balance between wind-driven pressure gradients and the Coriolis force on currents. The response diminishes away from the coast, returning to the balance between frictional and Coriolis terms, as in the classic Ekman model. On the contrary, transfer functions in the near-inertial frequency band show reduced magnitudes near the coast primarily due to friction, but exhibits the enhanced seaward response as a result of the inertial resonance. The idealized model simulations forced by local wind stress can identify the influences of remote wind stress and the biases in the data-derived transfer functions.

Appendices

Appendix A: Spatial scales of the surface wind field

The decorrelation length scales of the surface wind field off southern San Diego are examined with the COAMPS nowcast data with two spatial resolutions of 1.7 and 5.1 km (Hodur 1997). An example of spatial correlations of the hourly model wind at W in Fig. 1 for one year (2006) is shown in Fig 13. The length scales are asymmetric, i.e., relatively long seaward and short shoreward, in the range of 25 to 120 km.

Fig. 13

An example ofspatial correlations of the COAMPS hourly nowcast wind data at W (a white asterisk) in Fig. 1. a and c: ρ _{u u} (x,x ^′). b and d: ρ _{v v} (x,x ^′). a and b: 1.7 km resolution. b and d: 5.1 km resolution

Appendix B: Equilibrium of the model solutions

An example of model currents forced by low-frequency cross-shore wind stress (τ _x ) is presented with the hourly time series of u- and v-components at the center of a cross-shore line a in Fig. 1 and the surface layer (z=−0.5 m) over 30 days (Fig. 14a and b). The data for the first three days are excluded (a black box), and the rest of model solutions is used for the transfer function analysis and momentum budget analysis. Although the weak near-inertial oscillations are still imposed in the time series after three days, the model solutions at low frequency reach the equilibrium quickly.

Fig. 14

The hourly time series of model currents, forced by low-frequency wind stress in the cross-shore diction (τ _x ), at the center of a cross-shore line a in Fig. 1 and the surface layer (z=−0.5 m) over 30 days. The data for he first three days are excluded (a black box), and the rest of model solutions is used for the transfer function analysis. a u-component (cm s⁻¹), b v-component (cm s⁻¹)

Appendix C: Isotropic subsurface transfer function

From the isotropic relationship (4), the wind-coherent currents are as follows:

$$ \hat{\mathbf{u}}_{\mathrm{W}}(z, \sigma) = \mathbf{\mathsf{E}}(z,\sigma) \left[\hat{\uptau}_{x}(\sigma) + i\hat{\uptau}_{y}(\sigma)\right], $$

(24)

$$ {\kern3.1pc} = \left[ {\mathsf{E}}_{r}(z, \sigma) + i {\mathsf{E}}_{i}(z, \sigma)\right] \left[\hat{\uptau}_{x}(\sigma) + i\hat{\uptau}_{y}(\sigma)\right], $$

(25)

where E _r and E _i indicate the real and imaginary parts of the isotropic transfer function (E). The corresponding magnitude and argument are referred to as E and Θ^E, respectively, i.e.,

$$ {\kern.7pc} E = |\mathbf{\mathsf{E}}| = |\mathsf{E}_{r} + i\mathsf{E}_{i}|, $$

(26)

$$ {\Theta}^{E} = \tan^{-1} \frac{\mathsf{E}_{i}}{\mathsf{E}_{r}}, $$

(27)

The data-derived isotropic transfer function (E) is computed from the TJR wind stress and the detided HFR surface and ADCP subsurface currents (Fig. 15a and c). The model-derived isotropic transfer function (F) is based on the Ekman model (Fig. 15b and d) (e.g., Kim et al. 2009).

Fig. 15

Magnitudes and arguments of surface and subsurface isotropic transfer functions derived from HFRs and ADCP observations at T in Fig. 1 and the Ekman model. a E, b F, c Θ^E, and d Θ^F. f _c and δ _E denote the local inertial frequency (1.07 cpd) and the Ekman depth ($\delta _{E} = \pi \sqrt {2\nu /f_{\mathrm {c}}} = 8.705$ m, where ν=3×10⁻⁴ m ² s ⁻¹ is the kinematic viscosity), respectively

The peak near the inertial frequency decreases with depth, consistent with the model, penetrating up to about 8 m. However, the significant variance at low frequency does not appear in the model, which indicates the influence of pressure gradients near the coast because the Ekman model does not include pressure gradient terms (Fig. 15a and b) (e.g., Kim et al. 2009). The arguments of data- and model-derived transfer functions are very similar except for the sign shift at the zero frequency (Fig. 15c and d).

Appendix D: Time lag and veering angle

As the argument (Θ^E) of the isotropic transfer function (E in Eq. 24) contains the time lag (α) and veering angle (β), i.e., Θ^E=α+β (see Kim et al. (2009)),

$$ \hat{u}(\sigma) + i\hat{v}(\sigma) = E(\sigma)e^{i {\Theta}^{E}(\sigma)}\left[\hat{\uptau}_{x}(\sigma) + i\hat{\uptau}_{y}(\sigma)\right], $$

(28)

we can separate them with the relationship between individual components of wind stress and currents, i.e., anisotropic transfer functions.

As a simple case when ISCs are dominant over ASCs (e.g., offshore region), the anisotropic transfer functions can be paired (H _{x x} =H _{y y} and H _{x y} = −H _{y x} ) (e.g., Kim et al. 2009). The Fourier coefficients of currents are expressed with the isotropic convention:

$$ \hat{u} + i\hat{v} = \left(H_{xx}\hat{\uptau}_{x} + H_{xy}\hat{\uptau}_{y} \right) + i\left(H_{yx}\hat{\uptau}_{x} + H_{yy}\hat{\uptau}_{y}\right), $$

(29)

$$\begin{array}{@{}rcl@{}} {\kern2.2pc} &=& \left(|H_{xx}|e^{i{\Theta}_{xx}^{H}}\hat{\uptau}_{x} + |H_{xy}| e^{i{\Theta}_{xy}^{H}}\hat{\uptau}_{y}\right)\\ &&+ i\left(|H_{xy}|e^{i\left({\Theta}_{xy}^{H}+\pi\right)}\hat{\uptau}_{x} + |H_{xx}| e^{i{\Theta}_{xx}^{H}}\hat{\uptau}_{y}\right), \end{array} $$

(30)

$$ {\kern2.2pc} = \left(|H_{xx}|e^{i{\Theta}_{xx}^{H}} -i |H_{xy}|e^{i{\Theta}_{xy}^{H}}\right)\left(\hat{\uptau}_{x} + i\hat{\uptau}_{y}\right), $$

(31)

where H _{x x} has the zero veering angle and the time lag of ${\Theta }_{xx}^{H}$ ($\alpha = {\Theta }_{xx}^{H}$ and β=0^∘), and H _{x y} has the veering angle of −90^∘ and the time lag of ${\Theta }_{xy}^{H}$ ($\alpha = {\Theta }_{xy}^{H}$ and β=−90^∘). Thus, this comparison enables us to differentiate the veering angle and time lag when the argument is ambiguously computed under the isotropic assumption such as complex coherence and correlation.

At $\sigma = \sigma _{\mathrm {L}}, {\Theta }_{xx}^{H}$ is in the range of −2^∘ to −5^∘, and ${\Theta }_{xy}^{H}$ is in the range of −7^∘ to −8^∘ (Fig. 2m–p; Eq. 31). The estimated time lag (α) and veering angle (β) for three cases of the argument (Θ^E) at low frequency (Θ^E=−30^∘,−45^∘, and −60^∘) (e.g., Kim et al. 2010b) are shown in Table 1. For the case of σ=0 cpd (Ekman model), the time lang and veering angle are zero and −45^∘, respectively.

Table 1 Estimates of the time lag (α, degrees or hours) and veering angle (β, degrees) for three cases of the argument (Θ^E=α+β) of the isotropic transfer function at low frequency (σ=σ _L) using the pairs of time lags (${\Theta }_{xx}^{H}$ or ${\Theta }_{xy}^{H}$) and veering angles (0^∘ or −90^∘) of the anisotropic transfer functions. The negative time lag indicates that wind stress leads the currents, and the negative veering angle measures the amount of turn that the currents head to the right of the wind direction

At σ=σ _D, ${\Theta }_{xx}^{H}$ is in the range of 2^∘ to 6^∘, and ${\Theta }_{xy}^{H}$ is in the range of −61^∘ to −67^∘ (Fig. 7m–p). The estimated time lag (α) and veering angle (β) for four cases of the argument (Θ^E) at the diurnal frequency (Θ^E=−30^∘, −45^∘,−60^∘, and −75^∘) (e.g., Kim et al. 2010b) are shown in Table 2.

Table 2 Estimates of the time lag (α, degrees or hours) and veering angle (β, degrees) for four cases of the argument (Θ^E=α+β) of the isotropic transfer function at the diurnal frequency (σ=σ _D) using the pairs of time lags (${\Theta }_{xx}^{H}$ or ${\Theta }_{xy}^{H}$) and veering angles (0^∘ or −90^∘) of the anisotropic transfer functions. The meaning of the time lag and the convention of the veering angle are noted in Table 1

Appendix E: Estimates of ISCs and ASCs

Equations 14 and 15 are solved together frequency by frequency (e.g., Wunsch 1996; Kim 2009):

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{l} \hat{u}\\ \hat{v}\\ \end{array}\right] & = \left[\begin{array}{llll} \hat{\tau}_{x} & \hat{\tau}_{y} & \hat{\tau}_{x} & \hat{\tau}_{y} \\ \hat{\tau}_{y} & -\hat{\tau}_{x} & -\hat{\tau}_{y} & \hat{\tau}_{x}\\ \end{array}\right] \left[\begin{array}{l} \mathsf{I}_{d}\\ \mathsf{I}_{c}\\ \mathsf{A}_{d}\\ \mathsf{A}_{c}\\ \end{array}\right], \end{array} $$

(32)

where $\hat {u}, \hat {u}, \hat {\uptau }_{x}$, and $\hat {\uptau }_{y}$ are column vectors consisting of N sub-sampled Fourier coefficients at a single frequency (^‡ is the vector transpose), presented as follows:

$$ {\kern4pt} \hat{u} = \left[ \hat{u}_{1} \hat{u}_{2}\cdots \hat{u}_{N}\right]^{\dag},\\ $$

(33)

$$ {\kern4pt} \hat{v} = \left[ \hat{v}_{1} \hat{v}_{2}\cdots \hat{v}_{N}\right]^{\dag},\\ $$

(34)

$$ \hat{\uptau}_{x} = \left[ \hat{\uptau}_{x1} \hat{\uptau}_{x2}\cdots \hat{\uptau}_{xN}\right]^{\dag},\\ $$

(35)

$$ \hat{\uptau}_{y} = \left[ \hat{\uptau}_{y1} \hat{\uptau}_{y2}\cdots \hat{\uptau}_{yN}\right]^{\dag}. $$

(36)

Equation 1 is formulated as

$$ \mathbf{d} = \mathbf{Z}\:\mathbf{m}, $$

(37)

where d,Z, and m correspond to the currents, wind stress, and isotropy and anisotropy components. The estimated model coefficients ($\mathbf {\widehat {m}}$) are

$$ \mathbf{\widehat{m}} = \mathbf{P}\mathbf{Z}^{\dag}\left(\mathbf{Z}\mathbf{P}\mathbf{Z}^{\dag} + \mathbf{R}\right)^{-1}\mathbf{d}, \\ $$

(38)

$$ {\kern.7pc} = \left(\mathbf{Z}^{\dag}\mathbf{R}^{-1}\mathbf{Z}+\mathbf{P}^{-1}\right)^{-1}\mathbf{Z}^{\dag}\mathbf{R}^{-1}\mathbf{d}, $$

(39)

where P and R are the model and error covariance matrices (^‡ is the matrix transpose).

When the error covariance matrix (R) is replaced with an identity matrix (I), the model covariance matrix (P) will become a unique factor to adjust the inverse of snr (R P ⁻¹; β _k , k= 1, 2, 3, and 4), whose diagonal components reflect the contribution of each basis function:

$$\begin{array}{@{}rcl@{}} \mathbf{P} = \left[\begin{array}{llll} \beta_{1} & 0 &0 & 0\\ 0 & \beta_{2} & 0 & 0\\ 0 & 0 & \beta_{3}& 0 \\ 0 & 0 & 0 &\beta_{4}\\ \end{array}\right]. \end{array} $$

(40)

The inverse of snr for the estimates of ISCs and ASCs is assumed as 0.1, 0.1, 0.5, and 0.5 of the mean eigenvalues of the covariance matrix Fourier coefficients of wind stress, respectively, for I _d, I _c, A _d, and A _c at the given three frequencies (σ = σ _L, σ _D, and σ _f). The chosen ratios are less sensitive to the overall results.