1. Introduction

Accurate predictions of ice drift and currents for shelf and coastal regions are important for navigation, commercial fisheries, oil and gas exploration, climatic studies, and other human endeavor (Fowler et al. 2004; Leppäranta 2005). Because the wind is the major factor affecting oceanic motions along continental margins (Gill 1982; Wadhams 2000; Rigor and Wallace 2004), it is important to know the wind response in these regions as precisely as possible. Complicated topography, resonant and wave-trapping effects, and the formation of land-fast ice (for freezing areas) are among the factors that make continental margins among the most challenging areas for coastal ocean prediction research (Wang et al. 2003). At the same time, the availability of near-real-time wind observations for these areas allow effective application of regression models both for diagnostic and forecasting purposes (cf. Thorndike and Colony 1982; Fissel and Tang 1991).

The commonly used regression analysis of two vector series is based on the complex functional relationship between input and output vector series (cf. Greenan and Prinsenberg 1998); specifically,

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where V = (U, V) is the input vector series (wind), u = (u, υ) is the output vector series (ice drift or current velocity; herein “drift velocity”) and α = a + ib is a complex coefficient determined using a least squares regressional fit for the entire suite of wind and drift velocity observations. In our Cartesian coordinate system, u and U are positive to the east, and υ and V are positive to the north. This approach is widely used for the analysis of ice motions (cf. Thorndike and Colony 1982; Fissel and Tang 1991; Greenan and Prinsenberg 1998) and ocean currents (Cherniawsky et al. 2005).

Equation (1) also can be written

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where α₀ = |α| = a² + b² is the wind factor, and θ₀ = −arctan(b/a) is the turning angle of the drift velocity direction relative to the wind direction. The turning angle is measured clockwise (counterclockwise) to the wind in the Northern (Southern) Hemisphere. The specific values α₀ = 0.02 (a wind of 1 m s⁻¹ forces a 2 cm s⁻¹ ice drift) and θ₀ = 28°, which describe the relationship between wind and free ice drift in the open ocean, denote the “Nansen–Ekman ice drift law” (cf. Thorndike 1986; Wadhams 2000). Similar values are obtained for surface currents (α₀ = 0.01 − 0.04, θ₀ = 10°–40°). The problem is that Eqs. (1) and (2) describe an isotropic response of ocean motions to wind for which the parameters α₀ and θ₀ are invariant to the wind direction, ϕ. This requirement is counter to Overland and Pease (1988) who observed a strongly anisotropic response of ice drift to the wind near coastal boundaries. Similarly, Fissel and Tang (1991) established that proximity to the coast and direction of the wind relative to the coastline are major factors influencing variability in the wind factor and turning angle. To account for these effects, and to better define the dependence of α(ϕ) and θ(ϕ) on wind direction, we consider a two-dimensional (matrix) regression model based on regional winds. Such models are widely used in oceanography and meteorology for analysis and prediction of vector processes (cf. Emery and Thomson 2003).

The paper is organized as follows. In section 2, we examine the principal mathematical properties of our regressional model and estimate the response matrices and corresponding response ellipses (the drift velocity response to a unity wind velocity forcing). In section 3, we consider six different wind response cases beginning with the extreme case of a flat response ellipse near the coast (describing fully rectilinear motion) and ending with the other extreme of a circular ellipse (describing isotropic motion) in the open ocean. The model has been used to examine radar-measured ice drift motions on the northeastern shelf of Sakhalin Island in the Sea of Okhotsk and moored current meter data on the west coast of Canada. Results for the sea ice data only are reported in this study (section 4). Comparison of our four-parameter vector regression model and the more traditional two-parameter complex transfer function model for ice-drift data enables us to characterize the advantages and limitations of each model. The main results are discussed in section 5.

2. Vector regressional model

The effect of coastlines on winds and wind-driven currents in the coastal zone can be quantified using a two-dimensional vector model, which relates the wind V(t) and the free ice drift or current u(t) through the two-dimensional regression equation (cf. Cooley and Lohnes 1971; Maxwell 1977):

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Here

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and a_ij are the regression (response) coefficients linking the cross-shore (u) and alongshore (υ) components of drift velocity with the corresponding components (U, V) of the wind velocity; coefficients (ɛ_u, ɛ_υ) denote random noise. [Without loss of generality, we assume that both the drift velocity and wind have zero mean speed; thus, Eq. (3a) does not require a term representing the mean drift velocity.]

The coefficients a_ij in (3) are found through a least squares method that minimizes the expressions

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From (4) we obtain the following matrix relation (Cooley and Lohnes 1971):

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where 𝗔 is defined by expression (3b), 𝗗 is the autocorrelation matrix of the input variable (wind), and 𝗥 is the cross-correlation matrix between input and output variables:

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where the angle brackets 〈〉 denotes a time average. Terms involving random noise average to zero. From (5), it follows that

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where 𝗗⁻¹ is the inverse of 𝗗. Equation (6) defines the four response coefficients: a₁₁, a₁₂, a₂₁, and a₂₂. We note that, while the matrix 𝗗 is symmetric, the matrix 𝗥 is generally nonsymmetric and, consequently, the matrix 𝗔 is also generally nonsymmetric. This nonsymmetry complicates the wind–ice and wind–current relationships.

a. Response ellipses

The regression coefficients in model (3) define a response ellipse. This ellipse is the curve traced by the tip of the output response vector α = (α_u, α_υ) of the drift velocity through one complete rotation of a unity amplitude input wind vector (U₀, V₀) = ( sinϕ, cosϕ), where ϕ is the angle of the unity wind vector measured clockwise from north:

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According to (7), for each direction of the wind vector ϕ, there is a corresponding “wind factor” α, having relative drift speed α = |α(ϕ)| and direction ϕ = ϕ(ϕ). The angle between the drift velocity and wind vectors is the turning angle, θ = ϕ − ϕ. For the commonly specified case of an isotropic response, α_u = α_υ = const, θ = const, so that the response ellipse becomes a circle.

In general, each response ellipse is described by four invariant parameters: the semimajor (α = A_max) and semiminor (α = A_min) axes corresponding to the maximum and minimum ice drift (current) responses, respectively; the orientation of the semimajor axis (ϕ = ϕ_max); and the direction of the “effective wind” (ϕ = ϕ_max), the wind direction producing the maximum ice drift or current response. These four parameters are as follows.

• Direction of the “effective (noneffective) wind” ϕ_max (ϕ_min), the wind direction angle producing the maximum (minimum) drift response:

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• where ϕ_max (ϕ_min) is measured clockwise from north. Directions of ϕ_max and ϕ_min are related by ϕ_min = ϕ_max ± 90°.

• Semimajor ellipse axes:

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• Semiminor ellipse axes:

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• Orientation of the semimajor axis:

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The orientation of the semiminor axis is given by ϕ_min = ϕ_max ± 90°.

b. Eigenvectors of matrix 𝗔

If 𝗔 is a square matrix, V is a column vector, and λ is a scalar quantity (eigenvalue) such that

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where

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is the identity matrix, then V is said to be an eigenvector (latent vector) of the matrix 𝗔 (cf. Maxwell 1977). Each eigenvector is associated with a corresponding eigenvalue. The eigenvectors of the matrix 𝗔 determine the transformation from the vector V to the vector u. Because the directions of the eigenvectors are unchanged by this transformation (i.e., ϕ_λ = ϕ_λ), the eigenvectors give the direction along which the wind direction coincides with that of the wind-generated drift velocity. The eigenvalues, λ, are found from the characteristic equation (cf. Maxwell 1977):

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Equation (10) yields the quadratic equation

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which we use to define properties of the matrix 𝗔.

There are three invariants of the matrix 𝗔 determining the main properties of the transformation from wind to drift velocity (cf. Belyshev et al. 1983):

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The characteristic Eq. (11) has two roots (eigenvalues):

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whereby

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The directions of the eigenvectors are found from (10a):

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whereby

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If the matrix 𝗔 is symmetric (J₃ = 0), then for given eigenvalues λ₁ and λ₂, eigenvectors V₁ and V₂ are orthogonal and correspond to the principal ellipse axes associated with the maximum and minimum response (amplification) of the output series (drift velocity) relative to the input series (wind). However, if J₃ ≠ 0, then the eigenvectors of the response ellipses are nonorthogonal and are rotated relative to the principal ellipse axes. Depending on the sign of J₃, the turning angles θ have mainly positive or mainly negative values. The angle between two eigenvectors, which is equal to 90° if J₃ = 0, becomes smaller with increasing |J₃|. In the limit λ₁ = λ₂, there is only one eigenvector, and the turning angle always has the same sign except for the two zero-value points coincident with the eigenvector. The one-eigenvector condition (λ₁ = λ₂) may be presented as

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where D is the discriminant of Eq. (11). Case (16) is possible only if a₁₂ and a₂₁ have opposite signs. Moreover, if J²₁ < 4J₂, that is

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then Eq. (13) does not have real roots and the turning angle has always the same sign.

c. Asymptotic cases

The two limiting cases are the flat ellipse (rectilinear) response and the circle (isotropic) response.

1) Flat ellipse (one-dimensional motion)

This case occurs when the matrix 𝗔 is singular, that is, when the determinant

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A two-dimensional matrix has a zero determinant if one row (column) is proportional to the other row (column),

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where k is a constant. This case describes wind-induced rectilinear motions, whereby regardless of the wind direction, the ice (or water) moves back and forth along one direction only. Motions of this type are possible in a narrow channel or near the coast, where the wind-induced motions are restricted by the boundary. The magnitude and sign of k determine the direction of these motions. For example, if k = 0, then the direction is west–east; if k = 1, then the direction is southwest–northeast.

Using (8a) and (18), and several simple transformations, the directions of the most and least effective wind (i.e., the wind direction producing the maximum and minimum drift response, respectively) are found from the equation

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It is obvious from (7a) and (18) that when the first term in the brackets of (20) is equal to zero there is no oceanic response to the wind. In this case,

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whereby winds in the direction ϕ = ϕ₀ ± 180° produce no oceanic response; for any other direction, there is always a nonzero response. The direction of the maximum response may be found by setting the second term of (20) to zero:

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from which we obtain

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From (8b) and (23) it follows that the magnitude of the maximum response has the simple form,

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and that the corresponding direction of the maximum drift velocity depends only on coefficient k:

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More specifically, any drift motion in this case, including the maximum response, is in the direction ϕ = ±arctan(1/k). According to (13), the matrix (19) for this case has two eigenvectors:

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Note that, in contrast to the earlier cases considered above, the case of rectilinear motion is described by three, not four, independent parameters: a₁₁, a₁₂, and k (or ϕ_max, A_max, and ϕ_max).

2) Circular ellipse (isotropic response)

The second limiting case is the isotropic response, for which the drift response ellipse becomes a circle. This situation occurs when the matrix 𝗔 is antisymmetric (a₂₁ = −a₁₂) and the main diagonal coefficients are equal (a₂₂ = a₁₁); namely,

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This case is equivalent to case (1) with 𝗔 = α, whereby matrix (27) takes the form

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where α₀ = a²₁₁ + a²₁₂ and θ = arctan(a₁₂/a₁₁). Matrix (28) describes a combination of two elementary transformations, stretching and rotation, so that the isotropic response is described by only two independent parameters: a₁₁ and a₁₂ (or α₀ and θ). In general, matrix (27) has no eigenvectors. The sole exception is the case of isotropic response without turning (θ = 0), which occurs when a₁₂ = 0. In this case, wind vectors and response vectors always have the same directions and all vectors are the eigenvectors. The maximum turning angle θ = 90° (which is the same for all wind directions) corresponds to the case when the main diagonal elements of the matrix (27) are equal to zero (a₁₁ = 0).

3. Test examples

We now consider matrices 𝗔 whose parameters (regression coefficients a_ij) are representative of the six different cases of ice or current response to the wind discussed in the previous section:

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Here, the coefficients a_ij are expressed as the ratio of drift speed in CGS units to wind speed in MKS units [i.e., in cm s⁻¹ (m s⁻¹)⁻¹], corresponding to a percentage. Table 1 presents the derived matrix invariants, ellipse parameters, and eigenvector parameters for the six cases.

4. Ice drift on the Sakhalin shelf

To determine the ice drift response to the wind under observed oceanic conditions, we have used data collected in 1985–95 from coastal radar station “Odoptu” on the northeastern coast of Sakhalin Island in the Sea of Okhotsk (Pokrashenko et al. 1987; Shevchenko et al. 2004). The ice drift measurements were taken within 1-km circular areas at distances 4, 8, 12, and 16 km seaward of the coast. Present focus is on areas S1 and S4 (4 and 16 km, respectively). Ice drift vectors were determined three times per hour using major radar-reflecting ice flow targets located within radar coverage circles. Data were then interpolated and averaged into hourly time series of eastward (cross-shore, u) and northward (alongshore, υ) components of velocity. Local winds were measured simultaneously at hourly intervals at 10-m elevation. The year 1993 had the most extensive ice cover (Preller and Hogan 1998) and, therefore, the longest time series of ice drift. A continuous 73-day time series was collected during the period 12 March to 25 May 1993. The near-perfect agreement among estimated tidal motions from the various radar coverage circles signifies high-quality ice drift data. Tidal motions and mean drift were calculated and subtracted from the initial series.

Following removal of the tides, we next low-pass-filtered simultaneous series of wind and residual (detided) ice velocities to remove fluctuations with frequencies ≥1 cpd. The resulting low-frequency local wind and ice velocities were highly correlated. To quantify the influence of ice concentration on the ice drift response to wind, we then divided the data into four sequential 18-day segments characterized by distinctly different ice type and concentration (Tambovsky et al. 2001): 1) 12–30 March had an ice concentration of approximately 80%–90% with the ice field consisting of large and small broken floes; 2) 31 March–17 April had the highest ice concentration (95%–100%), consisting of large ice fields; 3) 18 April–6 May had reduced concentration (60%–80%) and diminished floe sizes; and 4) 7–25 May was associated with intensive melting with ice concentration reduced by ∼40%–50%. For each of the four time segments and two observational circles (S1 and S4), we computed the matrix 𝗔 for eight different cases:

• Location S1:

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• Location S4:

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where the matrix coefficients a_ij are in units of cm s⁻¹ (m s⁻¹)⁻¹, or percent.

Table 2 gives the matrix invariants, response ellipse parameters, and eigenvectors derived from expressions (7)–(28). For the three specific examples (cases S4-2, S4-3, and S4-4) presented in Figs. 7 and 8, the response ellipses are oriented along the coastline (ϕ_max = 9.5°–23.8° counterclockwise from the north). The pronounced flatness of the response ellipses indicates that the ice motions due to the wind are strongly anisotropic, with the ice response in the alongshore direction much greater than in the cross-shore direction (2.9%–6.1% versus 0.2%–1.9%, respectively). The alongshore values (2.6%–5.4%) of the response coefficients (the wind factor) are similar to those obtained by Fissel and Tang (1991) for the Newfoundland shelf. Response coefficients for the remote offshore observational area (S4) are greater by about 15%–20% than for the areas closest to shore (S1).

The above results reveal pronounced temporal changes in the ice drift response to the wind, apparently due to changes in ice properties. During the period of the highest ice concentration (period 2), the response ellipses are almost flat, indicating that the ice drift response was rectilinear (alongshore) (Fig. 7a). There are two eigenvalues, but the turning angles are mainly positive (Fig. 8a). In general, these ice-response ellipses resemble those for case C1 in section 3. For the early spring (period 1), and especially during the late spring (period 3), the response ellipses have larger magnitude and are more circular (Fig. 7b), indicating more intense cross-shore ice motions. Similarly, for the second period, the S1 and S4 matrices have two eigenvalues and a prevalence of positive turning angles (Fig. 8b). The response ellipses were of type C3. Finally, during the late spring (period 4), the response ellipses changed from flat to oval, similar to case C5 (Fig. 7c). For period 4, the matrices for both S1-4 and S4-4 had no eigenvalues and all turning angles were positive (Fig. 8c). According to our analysis, the last period was a time of free ice drift, while the three other periods were times of high internal ice stress and influence of the coast.

To compare the effectiveness of our vector regression model (3) versus the traditional complex transfer function model (1), we applied both models to the eight cases presented in Table 2 and estimated the residual (unexplained) variance. The results (Table 3) demonstrate that, for both offshore distances (S1, S4) and for the first three ice period segments (periods S1-1–S1-3 and S4-1–S4-3) (cases with substantial anisotropy of wind-induced ice motions), the vector regression model accounts for significantly greater variance than the traditional model. For example, for the strongest anisotropic case (period 2; Fig. 7a), the residual variances for the vector regression and traditional models have respective values: σ²_res(S1-2) = 28.5% and 57.2%; σ²_res(S4-2) = 17.9% and 54.2%. For our vector model, the residual variance becomes smaller for both velocity components (u, υ), while for the traditional model the residual variance of the cross-shore component (u) is increased relative to the initial value (Table 3). Thus, for markedly anisotropic ice motions, the complex transfer function model cannot accurately approximate the responses of both velocity components simultaneously, and performs well for highly energetic alongshore (υ) components of drift only. This contrasts with the last time segment (period 4), when the anisotropy was the smallest and the response ellipses were more circular (Fig. 7c), for which both models give almost identical results.

For the vector regression model, we define the mean wind factor α and the mean turning angle θ as

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Comparison of these parameters with those obtained for the traditional model (Table 3) indicates that model parameters are in reasonable agreement. However, the variance in α is significantly smaller than that of α₀ for the traditional model (1.88–2.41 and 1.62–3.12, respectively). For all eight cases, the “mean turning angle” (θ) was positive and larger than θ₀.

5. Summary and conclusions

Understanding the response of surface ice and currents to the wind is of critical importance in oceanography and marine engineering. The traditional approach for relating drift velocity to the wind is to assume a simple transfer function relationship involving two parameters: an amplitude variable that scales the drift speed to the wind speed and a directional variable that allows for an angular rotation of the drift velocity direction relative to the wind direction. Unfortunately, the two-parameter model, which was originally formulated to describe ice and current responses in the open ocean, is isotropic. We consider this unrealistic for coastal regions. In reality, how the ocean surface responds to the local winds depends on factors other than the wind, such as the orientation of the coastline and the regional bottom topography. For this reason, the assumption of isotropic response is likely invalid near the coast. To account for such effects, we have applied a two-dimensional (vector) regression model. In this model, the relationship between the wind and drift velocity (ice drift or current velocity) is described by four independent regression (response) coefficients, a_ij, linking the cross-shore (u) and alongshore (υ) components of the drift to the corresponding components (U, V) of the wind velocity. For each direction of the wind vector, ϕ, the method prescribes a “wind factor” α(ϕ) (relative drift speed) and “turning angle” θ(ϕ) (the angle between the drift velocity and wind vector).

Our description of the vector regressional model begins with the principal mathematical and physical properties of the model together with estimates of the response matrices and the corresponding response ellipses (ice or current velocity response to a unity wind velocity forcing). The major ellipse axes coincide with the direction of the “effective wind” (ϕ = ϕ_max) (the wind direction generating the strongest ice or current motions) while the minor axes of the response ellipse coincide with the direction of the “noneffective” wind (ϕ = ϕ_min) (the wind producing the weakest motions). In most cases, ellipses have corresponding eigenvectors. These eigenvectors denote the directions for which the wind and the induced drift motions are aligned. As a consequence, the eigenvectors separate zones, which have different signs for the turning angles, θ(ϕ). The angles are positive when the drift vector is directed to the right of the surface wind and negative when it is directed to the left of the wind. This anisotropy arises directly from the proximity to coastal boundaries. As we illustrate through the test examples in section 3, there are six different solutions (cases) for the matrix discriminant defined in section 2, ranging from a flat (rectilinear) response ellipse near the coast to an entirely circular response ellipse in the open ocean.

The vector regression model was applied to observed ice drift motions on the shelf of Sakhalin Island (section 4). Because the wind is the main factor determining low-frequency ice motions, the ice drift series is ideal for testing various models of wind-induced motions. These data not only have high signal/noise ratio, but they also indicate that the influence of other factors is relatively small (Wadhams 2000; Leppäranta 2005). The high-quality long-term ice drift data obtained in 1993 on the Sakhalin Island shelf (Shevchenko et al. 2004) enabled us to effectively estimate the efficiency of the four-parameter vector regression technique compared with the more traditional two-parameter complex transfer function approach. Estimates for different time periods and different offshore distances find that the vector regression model explains from 54% to 86% of the ice drift variance, while the traditional model only explains from 39% to 70% of the observed ice drift variance. For sea surface ice drift, ice concentration is a major factor influencing the response to the local wind field.

Because of its greater number of free coefficients, the four-parameter vector regression model should yield a smaller residual variance than the traditional two-parameter model. However, because the number of degrees of freedom in the dataset being analyzed decreases with an increase in the number of coefficients, calculation of the vector regression coefficients to the same level of confidence as the traditional model coefficients requires a slightly longer duration time series. The stability of the response ellipse parameters (relative to small changes in the parameters of the input functions) is the main criterion for determining the reliability of the results. Good agreement between the response ellipses for the Sakhalin shelf for two independent observation sites (S1 at 4 km offshore and S4 at 16 km offshore) and four different time periods (12 March–30 March, 31 March–17 April, 18 April–6 May, and 7 May–25 May 1993) indicates that the results in Tables 2 and 3 and Figs. 7 and 8 are highly reliable. It is also important to note that the structure of these ellipses have physical meaning in the sense that they account for the significant difference in ice drift response to alongshore and cross-shore winds. Our results confirm that an anisotropic, vector regression model is better for examining wind–ice and wind–current processes in coastal zone regions than an isotropic, complex transform model. Moreover, the vector regression model is more likely to capture surface dynamical features of the wind response than the traditional model.