1. Introduction

The nature of the oceans, as masses of fluid interacting with solid boundaries, makes possible, as a particular type of ocean–coast interaction, the phenomenon of an oceanic jet impinging on a coastal boundary. Among the many consequences produced by such an interaction, two features are of principal importance for the jet and its surroundings. These are its evident deflection by the coastal boundary and its division into two branches flowing along the coast. In the steady state, which is the case considered in this work, the phenomenon is observed only in specific oceanic regions having suitable boundary conditions, for instance at the exits of straits where a permanent outflow approaches the continental shelf. This specific oceanographic phenomenon is an example of the stagnation in line flows studied in fluid mechanics. A brief summary of these flows will help to define the objective of this work and to place it in perspective with other branches of that discipline.

Following the classic boundary-layer theory for the solution of the two-dimensional incompressible steady flow of a homogeneous fluid near a stagnation point at a rigid boundary, it is convenient to divide the flow into the potential (irrotational) and the viscous (with nonzero vorticity) regions (Fig. 1). The flow v_ψ in the irrotational region is known to be described by the streamfunction ψ = kxy, where k is a positive constant and v_ψ = −k × grad_hψ [see also other possible streamfunctions in, e.g., Peregrine (1981), Dorrepaal (1986), and Pedlosky (1987, p. 94)]. For the viscous region the governing Navier–Stokes equations can be reduced to an ordinary differential equation leading to similarity solutions. That solution was obtained numerically by Hiemenz (1911) and improved upon by Howarth (1935) [see, e.g., Batchelor (1970, chap. 5.5); Schlichting (1979) chap. V); Churchill (1988, chap. 14)]. Since that time, two-dimensional steady flow with a stagnation point has been referred to as “Hiemenz flow.” A similar division may be adopted in the oceanographic context. An oceanic jet approaching a continental slope has smaller turbulent eddy coefficients in the open ocean region than in the coastal region over the continental shelf. We will be primarily concerned with the changes experienced by such an oceanic free jet as it approaches the coast seaward of the 200-m isobath. Two important assumptions can therefore be justified for this situation. The first is that, as mentioned above, the flow is considered inviscid. The second is that the bottom depth is everywhere large enough to allow geostrophic velocities referred to a horizontal layer (e.g., at 200 m) of nearly zero motion to be computed from the density field. Unlike the theoretical jet studied by Hiemenz, oceanic jets cannot in general be considered irrotational. On the one hand, oceanic jets have large vertical shear and therefore a large horizontal component of relative vorticity. On the other hand, the vertical component of the relative vorticity even in straight impinging jets (i.e., ones with zero curvature vorticity) may be large due to the horizontal shear (shear vorticity).

The interest in quasi two-dimensional impinging flows having vorticity lies in the fact that the vorticity component parallel to the streamlines is susceptible to amplification by stretching and thus may be intensified near the boundary. The first studies of impinging two-dimensional flows with vorticity in the inviscid region were made in the context of the aeronautical sciences (Ferri and Libby 1954). Stuart (1959), after the studies of Li (1955, 1956) and Glauert (1957), considered the influence of uniform vorticity perpendicular to the plane of the two-dimensional flow of the fluid in the viscous region (such a fluid flow undergoes no stretching). Rott and Lenard (1959) and Kemp (1959) studied the same phenomenon but for axisymmetric stagnation flows. The effects of vorticity having a particular orientation susceptible to amplification by stretching when transported into the stagnation point boundary (turbulent) layer were described using a mathematical model by Sutera et al. (1963) and Sutera (1965). They demonstrated the possibility of vorticity amplification by stretching in quasi two-dimensional stagnation flow when the vorticity scale is larger than a certain neutral scale.

With the help of the above comments concerning general stagnation jets we are in a position to define this work as the study of the changes that a free jet with vorticity experiences when it approaches a solid boundary. Therefore, this work deals with the so-called external flow, that is, outside the viscous (turbulent) stagnation region, but such an external flow having vorticity. Furthermore, the geophysical context provides three new features that have not been addressed previously: namely, rotation, stratification, and a sloping boundary. To our knowledge the first work to address some consequences of the stagnation point region of a water jet in rotating fluid mechanics was that by Whitehead and Miller (1979) and later by Whitehead (1985a) in a laboratory simulation of the western Alboran gyre (WAG) in the Alboran Sea (Western Mediterranean) and the uplift near the Strait of Gibraltar. They pointed out that the growth of the gyre in their simulations may be due to the position of the stagnation point of the jet relative to coastal features at the point where the jet impinges on the African coast. They were unable to identify the existence of any theoretical or experimental studies of the properties of stagnation point flows in baroclinic, rotationally influenced jets. Whitehead considered the deflection by a vertical wall of a jet in a rotating fluid of constant density lying over a motionless fluid of slightly higher density. Using a momentum integral and considering a zero- as well as a constant-potential-vorticity jet, Whitehead was able to determine the relative size of the two branches of split flow as a function of the angle of incidence of the external jet.

The aim of this work is to characterize from observations the density and velocity changes experienced by a free, nearly geostrophic jet when it impinges on a sloping coastal boundary. We will therefore focus, as mentioned above, on the oceanic jet seaward of the continental shelf. We are aware that this topic is too wide and too complex to be resolved in one study. Consequently, our attempt is to describe the main hydrographic and kinematic properties that we have found to be clearly present in the impinging process. Therefore, the dynamical properties are excluded in this work. The specific case chosen in this study is the same one that motivated Whitehead’s studies, that is, the stagnation in line flow of the Atlantic current in the Alboran Sea. The description of the observations (section 2) is based on the density as well as on the vorticity and deformation fields. Due to the presence of a sloping coastal boundary, the changes in the jet’s structure occur in both the horizontal and vertical directions, that is, the changes are of a three-dimensional nature. Section 3 is devoted to the discussion and conclusions, and section 4 to the summary.

2. A case study: The Atlantic current in the Alboran Sea

The Atlantic current impinging on the African coast in the Alboran Sea is probably one of the best examples in nature of a quasi-steady jet impinging on a coast. Another possible example is the flow exiting the Tsugaru Strait north of Japan (Conlon 1982; Kawasaki and Sugimoto 1984). In both cases the geographic permanence of a jet is assured by a strait’s outflow. The knowledge of the physical oceanography of the Alboran Sea, although by no means complete, is one of the most extensive in the entire Mediterranean. The reader is referred to Parrilla and Kinder (1987) for a general description, Bucca and Kinder (1984) for meteorological effects, Gascard and Richez (1985) and Parrilla et al. (1986) for water masses, La Violette (1986) for short-term variability, Heburn and La Violette (1990) for gyre’s variability, Perkins et al. (1990) for the Atlantic inflow, Viúdez et al. (1996a, 1996b) for the gyre’s structure and ageostrophic motion, and Vázquez-Cuervo et al. (1996) for altimeter data analysis (and references therein).

The present data consists of 134 CTD stations (Δx ≈ 30 km, Δy ≈ 18 km) acquired in the Alboran Sea in September 1992 [described in detail in Viúdez et al. (1996a)]. Partial use is made of altimeter data [ERS1 (European Remote Sensing satellite) and TOPEX/Poseidon (ocean TOPography EXperiment)] in the discussion section in order to address the long-term temporal variability of the impinging process.

It has been the custom for ease of reference (e.g., La Violette 1986; Viúdez et al. 1996a) to use the term Atlantic jet (AJ) for the flow that, entering through the Strait of Gibraltar, usually impinges on the African coast at 3°40′W (Fig. 2). This flow then deflects to the left (facing downstream) and enters the eastern Alboran basin and then into the Algerian basin. The term WAG refers to the closed circulation including the flow that is deflected to the right at the stagnation point at the African coast. Strictly, the AJ and WAG are not two independent physical systems since there is water interchange between them (Lanoix 1974; Viúdez et al. 1996a). Since, in fact, they share a common density front, the current associated with this density front is here referred to as the impinging current, impinging jet, or simply the current, in order to distinguish it from the AJ (the impinging current being wider than the AJ). Using this terminology, the current impinges on the sloping African coast and is deflected and divided into two branches: the AJ, or left branch, and the WAG current, or right branch. Note that the horizontal and vertical scales of the African continental shelf in the impinging region (∼15 km and 200 m, respectively; see, e.g., Fig. 3) are of the same order as the horizontal and vertical scales of the Atlantic jet (or the impinging current). Thus, we expect that the sloping of the shelf influences the impingment process. Furthermore, since the width of the shelf is comparable to the meridional distance between CTD stations (∼18 km), we expect also that this influence may be detectable in the hydrographic data. Other possible features associated with the impinging process, but having spatial scales smaller than the spacing between CTD stations, could be missed or aliased with the present data resolution.

b. Kinematics

In this section we show the main kinematic characteristics of the impinging region based on the vorticity and deformation fields. As part of the density data postprocessing, we obtained an adjusted, dynamically balanced three-dimensional velocity field (Viúdez et al. 1996b) through the dynamical assimilation of the density data into a PE model (Haney 1974, 1985) using the digital filter initialization technique (Lynch and Huang 1992). This method provided a dynamically balanced density and three-dimensional velocity field at the grid points of a 50 × 40 × 48 regular matrix (δx ≈ 9.3 km, δy ≈ 6.7 km, δz = 4 m) covering the entire Alboran Sea. This assimilation procedure, although not essential for most of the results presented here, provides a more accurate estimate of the velocity field than geostrophic.

The notation used in the remainder of the paper follows that in Gurtin (1981) and Truesdell (1991). We have adopted this notation instead of the Gibbsian dyadic notation (e.g., Morse and Feshbach 1953; Godske et al. 1957; Tai 1992), more common in hydrodynamics, because it is the one most commonly used in continuum mechanics, especially in the study of deformation. In particular a tensor S is considered to be a linear map that assigns to each vector b a vector Sb. The spatial gradient, curl, and divergence are denoted as grad, curl, and div. The advective derivative of a vector b is denoted as (grad b)v, where v is the velocity field. Directional or intrinsic derivatives of a scalar ϕ or vector-valued function b in the direction of a unit vector s are denoted as δϕ/δs ≡ s·gradϕ and δb/δs ≡ (grad b)s, respectively. These operators are coordinate independent. Partial derivatives with respect to Cartesian coordinates {x_i}, i = 1, 2, 3, are denoted as ϕ_,i, ≡ ∂ϕ/∂x_i. The tensor product a ⊗ b of two vectors a and b is the tensor that assigns to each vector u the vector (b·u)a [that is, (a ⊗ b)u = (b·u)a]. Given a Cartesian coordinate frame {e_i} = {i, j, k} the components S_ij of a tensor S are defined by S_ij ≡ e_i·Se_j, so that S = S_ije_i ⊗ e_j. In particular, grad a = a_i,je_i ⊗ e_j. Latin indices i, j, k ∈ {1, 2, 3}, while Greek indices α, β ∈ {1, 2}. Summation convection is implicit in repeated indices, except for caret indices (e.g., α̂α̂ represents 1 1 or 2 2). The trace is the linear operation satisfying tr(a ⊗ b) = a·b. The transpose S^T of S satisfies Sa·b = a·S^Tb; in particular (a ⊗ b)^T = b ⊗ a. The inner product of two tensors S and T is defined as S·T ≡ tr(S^TT), or in componentsS·T = S_ijT_ij. The magnitude |S| of the tensor S is defined as |S| ≡ (S·S)^1/2.

1) Curvatures and backing

Due to the different scales in the horizontal and vertical components of the velocity field v, let us decompose it into perpendicular horizontal and vertical vectors

vxv_hυ₃kv_hk(1)

Defining director vectors for the 3D flow s(x) ≡ v/υ (for υ ≠ 0) and for the horizontal flow s_h(x) ≡ v_h/υ_h (for υ_h ≠ 0), we have the relation for the director vectors:

υsυ_hs_hυ₃k(2)

Let n and b be the normal and binormal unit vectors, respectively, of the streamlines of the 3D flow, and n_h ≡ k × s_h the normal unit vector of the horizontal flow. Then the intrinsic trihedron (s, n, b) and the trihedron (s_h, n_h, k) are a set of orthogonal measuring vectors, in general anholonomic (e.g, see Schouten 1954; Marris and Passman 1968; Gola̧b 1974, chap. IV), for the 3D flow. The Frenet–Serret-type relations for the vectors (s_h, n_h) are

View Expanded

which may serve as a definition of the streamline curvature of the s_h and n_h lines, here denoted Θ_S and Θ_N, respectively, and for the rotation of s_h along the k line, or backing, here denoted Θ_Z. The formulas on the rhs of (3) are just a 90° counterclockwise rotation [k × ()] of the ones on the lhs. Note that curvatures may be computed in an equivalent way from Θ_S = k·curl s_h and Θ_N = k·curl n_h. The spatial fields of curvatures and backing for the impinging current are represented in Fig. 7. The streamline curvature field Θ_S(x), consistent with the pressure anomalies contours, is negative to the right (facing downstream) and positive to the left of the impinging current, where the circulation is anticyclonic and cyclonic, respectively. The curvature of the n_h lines, Θ_N, can be considered as a measure of the divergence of s_h(x), or diffluence of the s_h lines, since Θ_N = k·curl n_h = div s_h. The Θ_N(x) field shows diffluence (Θ_N > 0) upstream of the impinging region, implying therefore direction divergence (υ_hΘ_N > 0), but confluence (Θ_N < 0) past (i.e., after the flow division) the impinging region, and therefore implying direction convergence (υ_hΘ_N < 0). Since the horizontal flow is almost nondivergent [recall that div v_h = div(υ_hs_h) =δυ_h/δs_h + υ_hΘ_N], the previous pattern of direction divergence is completed with speed convergence (δυ_h/δs_h < 0) upstream and speed divergence (δυ_h/δs_h > 0) downstream of the impinging region (Fig. 8).

The backing Θ_Z is a measure of the change in direction of the velocity field along the vertical. The backing distribution (Fig. 7c) shows that Θ_Z < 0 upstream far from the impinging region, but Θ_Z > 0 upstream, and Θ_Z < 0 past the stagnation point. Among these different areas characterized by different signs in Θ_Z, probably the most relevant one is where Θ_Z < 0, far upstream, because of the large velocities in the AJ. This importance can be easily visualized by looking at the streamlines at different depths in the AJ (Fig. 9). It is clearly observed how deep streamlines, “initially” aligned in the vertical with the surface ones in the AJ, are deflected to the left (facing downstream) “before” the surface ones.

In meteorology and oceanography the backing Θ^g_Z of the geostrophic velocity v^g is important because it represents the advection of density by the geostrophic flow. By defining the vertical shear of the geostrophic velocity through the “thermal wind” relation v^g_,3 ≡ −k × grad_hb with b ≡ gρ(fρ_o)⁻¹, and introducing the unit basis vectors (s^g, n^g) for the geostrophic flow, s^g ≡ v^g/υ^g and n^g ≡ k × s^g, we find that v^g_,3 = (υ^g_,3s^g)_,3 = υ^g_,3s^g + υ^gΘ^g_Z n^g, and the s^g component of the thermal wind relation is −υ^gΘ^g_Z = δb/δs^g. Therefore Θ^g_Z > 0 implies δb/δs^g < 0. Since the geostrophic advection of b is given by −v^g·grad_hb = −υ^gδb/δs^g, the above condition implies positive geostrophic advection of density, or in other words, a decrease of density along the geostrophic streamlines.

2) Vorticity

In the (s, n, b) reference frame the vorticity is expressed as

View Expanded

(Masotti 1927; Bjørgum 1951), where Ω_s and κ_s, the abnormality and curvature of the s line, are quantities independent of the magnitude of the velocity field. Note that what is frequently called secondary vorticity (ζ·s/υ) is, in fact, the abnormality Ω_s (e.g., Truesdell and Toupin 1960, p. 386), which is a measure of the departure of v from the property of having a congruence of surfaces with s as a normal (e.g., Ericksen 1960, p. 819). In the (s_h, n_h, k) reference frame the relative vorticity may be expressed as

View Expanded

where ζ_ph is the horizontal component of the pseudovorticity ζ_p ≡ ζ_ph + ζ₃k (Hoskins 1975; Davies-Jones 1991). Horizontal pseudovorticity is, in fact, a +90° rotation of the vertical shear

v_h,3υ_hs_h,3υ_h,3s_hυ_hZn_h(6)

and then

ζ_phkv_h,3υ_hZs_hυ_h,3n_h(7)

in the (s_h, n_h) reference frame. In (5) ζ is decomposed in horizontal pseudovorticity, the vertical component of vorticity ζ₃, and vorticity due to the vertical velocity. The component ζ₃ in (5) is itself decomposed into curvature vorticity (υ_hΘ_S) and shear vorticity (−δυ_h/δn_h).

Relative streamwise vorticity (ζ·s), or secondary vorticity (ζ·s/υ), or helicity (ζ·v), and cross-stream vorticity (ζ·n), are frequently used in the description of flows (Hawthorne 1951; Hawthorne and Martin 1955; Squire and Winter 1951; Marris 1964; Lakshminarayana and Horlock 1973; Scorer 1978, chap. 3). More recent research dealing with the influence of the cross-stream vorticity of the external flow on the stagnation-point flow and vorticity amplification theory can be found in Sadeh and Brauer (1980) and Dhanak and Stuart (1995). The explicit use of streamwise vorticity in the study of atmospheric or oceanic phenomena appears to be more limited. However, since streamwise vorticity is closely related to the backing/veering of v_h along k, it has received, at least implicitly, considerable attention in both atmospheric and oceanic processes. Scorer and Wilson (1963) uses it to study the instability in atmospheric gravity waves. Davies-Jones (1984) and Brandes et al. (1988) consider the storm-relative streamwise vorticity to play a main role in the origin of organized rotation in supercell storms. Evidently, none of the above quantities [ζ·s, ζ·n, ζ·s/υ, ζ·v, mean helicity 〈ζ·v〉 (Speziale 1987), etc.] are Galilean invariant. In Brandes et al. (1988) it is argued that only streamwise vorticity relative to the reference system in which the storm is steady has physical significance. On the other hand, most of the previous studies deal with steady-state circulation, which also is not a Galilean invariant concept, and therefore they also assume a preferred reference frame (precisely, the reference frame in which the flow is steady). Dealing also with boundary interaction processes there is also a preferred reference system, that is, that in which the boundary is at rest. This is implicitly assumed when the motion is referred to rotating (fixed to the earth) coordinate axes. Therefore, streamwise vorticity can be considered a suitable quantity in the kinematic description of impinging jet processes. Using (2) and (5) the streamwise relative vorticity can be written as

View Expanded

Since υ₃/υ = O(10⁻³) the analysis of streamwise vorticity ζ·s = υΩ_s is, to a good approximation, the analysis of the component of the horizontal pseudovorticity in the direction of the horizontal flow ζ_ph·s_h = −υ_hΘ_Z. On the other hand, the n_h component of the vorticity

View Expanded

is approximately the n_h component of the pseudovorticity, which from (7) is the vertical derivative of the horizontal speed ζ_ph·n_h =υ_h,3. The vorticity components in the (s_h, n_h, k) directions for the impinging current are shown in Fig. 10. It is observed that cross-stream vorticity experiences a large decrease from 500 to 200 (×10⁻⁵ s⁻¹) when the impinging current approaches the coast. The vertical component ζ₃, having contours quite parallel to the isobars, is relatively conserved along streamlines in the impinging area. Note that (ζ_ph·n_h)/ζ₃ = O(5 × 10²). Streamwise vorticity, however, decreases from ∼0 to −150 × 10⁻⁵ s⁻¹ as the current approaches the coast. Streamwise vorticity of a water parcel is therefore the only vorticity component that, in spite of the decrease in speed (Fig. 11), increases in magnitude as the fluid element impinges on the coast. As a water parcel approaches the coast the horizontal pseudovorticity ζ_ph, initially almost pointing eastward, experiences a large contraction together with a counterclockwise rotation in such a way that the magnitude of the streamwise vorticity becomes comparable to the magnitude of the cross-stream vorticity. Since ζ_ph·s_h = −υ_hΘ_Z, veering (Θ_Z < 0, clockwise rotation of v_h with k) always implies positive streamwise pseudovorticity, and, conversely, backing (Θ_Z < 0, counterclockwise rotation of v_h with k) always implies negative streamwise pseudovorticity. In steady state, and neglecting vertical advection, the material derivative of ζ_ph·s_h is given by the horizontal alongstream derivative of ζ_ph·s_h,

View Expanded

which states that a decrease of negative ζ_ph·s_h (Θ_Z > 0) along streamlines may be due to a speed divergence (δυ_h/δs_h > 0) or to a streamwise increase of the backing Θ_Z. Since the region upstream of the stagnation point is characterized by speed convergence (δυ_h/δs_h < 0), there is a steady decrease of ζ_ph·s_h when there is a streamwise increase in the backing (δΘ_Z/δs_h) and when this increase in backing is such that υ_hδΘ_Z/δs_h > −Θ_Zδυ_h/δs_h. The streamwise increase of the backing upstream of the stagnation area can be easily deduced from the Θ_Z distribution (Fig. 7c). The corresponding distribution of −υ_hδΘ_Z/δs_h (Fig. 12) shows that the alongstream changes in ζ_ph·s_h are, in fact, due to the changes in the backing. The distribution of−Θ_Zδυ_h/δs_h (not shown) has positive values upstream of the stagnation area (because Θ_Z > 0 and δυ_h/δs_h < 0) but these values are negligible with respect to the values of −υ_hδΘ_Z/δs_h. These distributions show, therefore, that the streamwise changes in the horizontal pseudovorticity are, in fact, due to the corresponding streamwise changes in backing.

The kinematic properties presented above are significant for several reasons. First, they represent a quantitative description of the three-dimensional velocity field in the impinging region. As such, they provide quantitative and meanful data for testing theories or models of the impinging process. One could, in principle, have done this description directly by simply mapping the velocity at different levels. Such a method, however, would be largely imprecise and only qualitative in nature. From the already mentioned non-Galilean invariance of these kinematic quantities, it is clear that they have no dynamical significance. Since the impinging jet is characterized by large changes of opposite sign in speed divergence and direction divergence (the changes in horizontal divergence can be neglected in comparison with the changes in speed divergence or direction divergence), the impinging process could be considered a very good example of a conversion between speed divergence and direction divergence and be studied using the interchange equations proposed by Hollmann (1958). However, such a treatment would also be considered a description and not an explanation of the changes in speed divergence and direction divergence (see equivalent conclusions regarding the shear and curvature vorticity equations in Viúdez and Haney 1996). The kinematic quantities presented above are also important because of their connection to certain experimental methods. For example, a satellite thermal image may be used to infer the streamline curvature field but, in general, not the velocity field. From the trajectory of a single isopycnal float we can measure the curvature vorticity, but not vorticity itself. We will see in the next two sections that many of the properties of the velocity field in this study, including the deformation, can be deduced from the single fact that the acceleration field is divergent in the impinging region.

3) Rotation and deformation

The previous remarks have shown that the relative vorticity of a water parcel in the impinging current experiences notable changes as it approaches the African coast. However, since any continuous motion at a given instant of time is composed of a translation, a rigid rotation, and a pure deformation, it becomes equally important to determine the relative significance of the deformation field in the stagnation current. A measure of the ratio of the magnitude of rotation to the magnitude of deformation was introduced by Truesdell (1953, 1954) by defining the kinematical vorticity number as the dimensionless scalar

View Expanded

where D and W are the symmetric and skew-symmetric parts of the velocity gradient (the stretching and spin tensors, respectively)

View Expanded

Note that |W|² = W·W = W_ijW_ij = (1/2)ζ², where ζ ≡ ‖curl v‖ is the magnitude of the relative vorticity. Since a rigid rotation has D_ij = 0, ζ ≠ 0, and a nonrigid irrotational motion has ζ = 0, D·D ≠ 0, the kinematical vorticity number provides a numerical degree of rotationality for all possible motions (exception are the rigid translations) on a scale from 0 (rigid rotational motion) to ∞ (nonrigid irrotational motion). When W_K = 1, spin and stretching are precisely balanced. Since

View Expanded

we have

View Expanded

and therefore from (11)

View Expanded

Since in the impinging current (1/2)ζ² = O(10⁻⁵ s⁻²) and L·L^T = O(10⁻¹⁰ s⁻²) at, say 100 m, the kinematical vorticity number is W_K = 1 ± O(10⁻⁵). The W_K distribution is shown in Fig. 13 as (W_K − 1) × 10⁵. It is observed that, while the magnitude of rotation is larger than the magnitude of deformation in the WAG (W_K > 1), the magnitude of deformation exceeds the magnitude of rotation when the current impinges on the African coast (W_K < 1). Note that in the impinging region W_K ≪ 0 from the surface to ∼150 m, while in the AJ this only occurs in the surface (from 0 to 50 m) layer. Because of this clear signature, deformation is a very important quantity in the kinematic description of impinging jet processes. Since ζ² is a positive quantity, the positive or negative departure of W_K from 1 depends on the sign of L·L^T. IfL·L^T > 0 then W_K < 1. The quantity L·L^T = L_ijL_ji = υ_i,jυ_j,i at 100 m is shown in Fig. 14. It exhibits positive values in the impinging area and negative values in the WAG area, according to the W_K distribution. The fact that we are considering nondivergent motions (div v = 0) allows us to give an alternative kinematic interpretation of the term L·L^T. By taking the divergence of the advective derivative Lv we have

LvvvLL^T(16)

or in Cartesian components

υ_i,jυ_j,iυ_i,jiυ_jυ_i,jυ_j,i(17)

The first term on the right-hand side is zero since div v = υ_i,i = 0. Therefore, L·L^T represents the divergence of the advective derivative of v for nondivergent motions. Since the acceleration field a = v_,t + Lv, the term L·L^T also represents the divergence of the acceleration field (div a) for steady flows (v_,t = 0). Since (a·k)_,3 is negligible for hydrostatic flows, we have for nondivergent hydrostatic steady flows div a ≈ div a_h = div(grad v_h)v, where grad v_h = υ_α,i e_α ⊗ e_i. The advective derivative of v_h is shown in Fig. 15. Besides the centripetal acceleration of the fluid elements in the WAG circulation directed toward its center, the deceleration of the current before impinging on the coast (a_h vectors are opposite to the flow direction) and the tendency of the flow to accelerate after the flow division (a_h vectors have a positive component in the direction of the flow) are also clearly observed. In the steady state, this deceleration–acceleration of the fluid produces the divergence of a_h (div a_h > 0), which implies L·L^T > 0 and therefore W_K < 1.

It should be noted that, as it is usual practice in oceanography, the velocity field used in the computation of W_K is referenced to the rotating earth. Thus, W_K is in fact a relative vorticity number. If velocity is instead referenced to a nonrotating system absolute vorticity should replace the relative vorticity in (11). The absolute vorticity number so obtained is one order of magnitude larger than the relative W_K and is always positive in the study domain, with a relative extremum, in this case a maximum, in the impinging region. This shows the large effect that the solid (nondeformable) coastal boundaries have in communicating their own rotation to the oceanic flow.

Separation between horizontal and vertical velocity components for the term L·L^T may be clarified by using (1) to write

View Expanded

Defining L_h = grad_hv_h ≡ υ_α,βe_α ⊗ e_β and using the identity (a ⊗ b)·(c ⊗ d) = a·c b·d, we obtain

LL^TL_hL^T_hv_h,3υ₃υ²_3,3(19)

The horizontal version of (16) is

L_hv_hv_hv_hL_hL^T_h(20)

using div v_h = −υ_3,3, and moving L_h·L^T_h to the lhs, we get

L_hL^T_hL_hv_hv_hυ_3,3(21)

Thus, L_h·L^T_h represents the horizontal divergence of the horizontal advective derivative of v_h when the flow is horizontal and nondivergent. Note also that, if υ₃ = 0, then by (19) L·L^T = L_h·L^T_h. These results have to be taken into account since in our study υ₃/υ_h = (10⁻³). Therefore, the distributions of L·L^T in the impinging current (Fig. 14) and div(L_hv_h) are mainly due to the horizontal flow v_h sinceL·L^T ≈ L_h·L^T_h ≈ div(L_hv_h) when υ₃/υ_h ≪ 1.

4) Stretching

The study of stretching may be accomplished by means of the principal stretchings and principal axes of stretching. The reader is referred to Chadwick (1976), Gurtin (1981), and Truesdell (1991) for a detailed exposition of these topics. The principal stretchings are the proper numbers of the symmetric tensor D. Briefly, and practically quoting Chadwick (1976, p. 75–76), since D is a symmetric tensor, it possesses three proper numbers called the principal stretchings, and an orthogonal triad of proper vectors d_k(x, t), k = 1, 2, 3 whose directions are called the principal axes of stretching. The principal stretchings (at a point x) are the rates of extension per unit length experienced by material line elements currently aligned with the principal axes of stretching. Moreover, the rates of change of the angles between these lines are momentarily zero. Material elements along these principal axes of stretching are instantaneously suffering no rotation with respect to one another. Their motion as lines, no account being taken of the motion of particles along them, is instantaneously rigid; hence the spin is the angular velocity of the principal axes of stretching, in the ordinary sense of rigid motions (Truesdell and Toupin 1960, p. 355). The relative motion in the vicinity of the point x consists of a triaxial stretching, represented by D, superimposed upon a rigid rotation specified by W. Since it can be shown that the stretching tensor D is dominated by the large values of v_h,3, a direct representation of the stretching axes of the 3D flow is, in general, unsuitable (see the appendix).

We may instead consider the deformation of the horizontal two-dimensional flow. In this representation the 3D flow is thought to be composed of a set of 2D horizontal layers, and vertical motion is not considered. In this description every layer is two-dimensional and is treated from a kinematic point of view as an independent material system. The corresponding velocity gradient is L_h (see the appendix for symbol definitions) and the stretching tensor is D_h. The proper unit vectors c_α and the proper values λ_α satisfy

D_hc_αλ_α̂c_α̂(22)

The spectral decomposition of D_h is D_h = λ_αc_α ⊗ c_α. Since div v = div v_h + υ_3,3 = 0, then div v_h = tr(D_h) = Σ λ_α = −υ_3,3. But υ_3,3/υ_h = O(10⁻³) and therefore the proper values λ_α are to a large extent equal in magnitude but opposite in sign. The spatial field of proper vectors c_α in the impinging current is shown in Fig. 16. The direction of these vectors indicates therefore the stretching axes of the horizontal 2D flow. Since the sense of the vectors is irrelevant (−c_α are also proper vectors), we have adopted the convention of representing the vector ±λ_α̂c_α̂ satisfying that sgn(±λ_α̂c_α̂·v_h) is equal to sgn(λ_α). Therefore, in Fig. 16 when λ_α̂c_α̂ is in the direction of v_h, there is positive stretching (λ_α > 0) in that direction. It is observed that the stretching axes ±c_α of the current before impinging on the coast form an approximately 45° angle with the flow. Therefore, the magnitude of rotation equals the magnitude of deformation, a result already observed from the kinematical vorticity number distribution. As a water parcel in the current approaches the coast (we are assuming steady state in this description) a rotation (relative to the flow direction) of its stretching axes occurs in such a way that the axes become almost parallel and normal to the coast in the impinging region. There occurs an enhancement of the stretching, with positive stretching in the direction parallel to the coast and negative stretching in the direction normal to the coast.

3. Discussion and conclusions

The results concerning the impinging process may be summarized by the sketch in Fig. 17. In the horizontal we distinguish between several areas labeled from I to IV. The first area (I) is upstream, far from the impinging region. In this area the horizontal flow is not normal to the coast, the stagnation streamlines have relatively large horizontal curvature, and the backing is negative (Θ_Z < 0). When the flow approaches the coast and arrives at the impinging region proper, there is an upstream impinging region (area II) where the flow has speed convergence (δυ_h/δ_h < 0), and therefore diffluence (Θ_N > 0) and speed deceleration (dυ_h/dt ≈ υ_hδυ_h/δs_h < 0 in the steady state). This region is characterized by positive backing (Θ_Z > 0) and, therefore, by negative streamwise pseudovorticity (−υ_hΘ_Z < 0). Past the impinging region, the two horizontal flow branches (areas III and IV) experience speed divergence (δυ_h/δs_h > O), therefore confluence (Θ_N < 0) and speed acceleration (dυ_h/dt ≈ υ_hδυ_h/δs_h > 0 in the steady state). These two areas (III and IV) are characterized by negative backing (Θ_Z < 0), and therefore by positive streamwise pseudovorticity (−υ_hΘ_Z > 0). While in area I the magnitude of the deformation and rotation are similar (W_K ≈ 1), in areas II, III, and IV, which correspond to the upstream and past the stagnation point regions, the magnitude of deformation exceeds the magnitude of rotation (W_K < 1). This fact is related to the divergence of the acceleration field. As the current approaches the coast there occurs a rotation (relative to the flow direction) of the stretching axes of the fluid element in such a way that they become almost parallel and normal to the coast in the impinging region.

The main general result of the present study is that, if it were not clear enough, there is an interaction between the current and the coast; that is, the current really impinges on the coast (for a rotating observer) and is deflected by the coast. Whereas, after looking at the satellite image (Fig. 2) and the density distribution (Fig. 3), our intuition tells us that this interaction is evident:the quantitative analysis of vorticity and deformation that we have done is the objective way of expressing this fact and describing how this coast–jet interaction takes place. This impinging process pattern in the Alboran Sea, while it appears to be the most frequent one, is not the only one possible. An interesting different case was observed by García-Lafuente et al. (1997), in which the AJ was “prematurely” deflected to the east by the presence of a small-scale cyclonic eddy located in the impinging region.

To first order the AJ impinging process is considered a quasi-steady process. That is, the temporal scale associated with the water elements passing through the impinging region (about one day) is considered to be small compared to the temporal scale of the impinging process itself. Probably the shorter periodic phenomenon that can substantially affect the impinging process is the quasiperiodic 9-day oscillation of the AJ reported by Perkins et al. (1990). Large-scale low-frequency variability in the Alboran Sea has been studied only from remote sensing data (Heburn and La Violette 1990; Vázquez-Cuervo et al. 1996). A time sequence of a part of the semiannual cycle of residual sea level in the Alboran Sea is shown in Fig. 18. The semiannual cycle was calculated by fitting at each grid point a semiannual harmonic to the merged TOPEX/Poseidon ERS1 altimeter data from October 1992 to December 1993. This dataset was processed by fitting the TOPEX/Poseidon orbit to the ERS1 orbit [which reduces substantially the orbit error of the ERS1 data; see Ayoub et al. (1997) for more information on the data processing]. It was found that in the Alboran Sea the annual and semiannual cycle each account for nearly 50% of the resolved variability. Clearly visible is an eastward propagation of the semiannual wave. It therefore appears that the impinging jet process could also have relevant annual and semiannual components. However, it is important to note that the total amplitude of the temporal variability of sea level (∼10 cm) is smaller than the mean sea level anomaly in the WAG–AJ system (∼25 cm, e.g., Perkins et al. 1990). Thus, for example, the meridional currents at the coast near 4°W are always directed toward the coast (southward) even though the disturbance currents inferred from the sea level gradients in Fig. 18 show a reversal.

Some of the experimental results shown here for the 2D flow are in qualitative agreement with the characteristics of the flow v_ψ ≡ −k × grad_hψ described by the streamfunction ψ ≡ kxy (Fig. 1). The field v_ψ is nondivergent, and the speed divergence δυ_ψ/δs_ψ = (x² − y²)/(x² + y²) is zero when x = y, negative when |x| < |y| and positive otherwise, which qualitatively corresponds to the speed-divergence pattern shown in Fig. 17 for areas II, III, and IV. Since L_ψ ≡ grad v_ψ = k(i ⊗ i − j ⊗ j), the directions i and j have been chosen to be the principal axes of stretching axes of the flow v_ψ, and D_ψ = L_ψ, W_ψ = 0. However, this 2D streamfunction cannot account for the 3D characteristics (z dependence) of the impinging process found in this study. Considering v_ψ a model of the 2D circulation, where vertical motion is not taken into account, the 3D flow is described by a set of streamfunctions ψ_i, for i = 1, . . . , N, with N being the number of vertical levels. In such a flow description the property of irrotationally of every v_ψi may be judged by considering the relative importance of the vorticity term in the horizontal acceleration field of the impinging current. We therefore interpret that the relative rotation is negligible if the relative vorticity term of the relative acceleration field may be neglected (Truesdell 1953). This analysis is done by using the Lagrange formula for the 2D acceleration field

View Expanded

A measure of the relative importance of the vorticity can therefore be obtained using the ratio

View Expanded

which is a 2D version of the dynamical vorticity number introduced by Truesdell (1953, 1954). Note that W_D is a purely kinematic quantity and that, unlike W_K, is not Galilean invariant. The value W_D gives an indication of the importance of the rotational term in the acceleration field as written in one particular frame. In the steady state, W_D represents the ratio between the vertical component of the relative vorticity and the magnitude of the speed gradient (|ζ₃|/‖grad_hυ_h‖). The distributions of ζ₃, ‖grad_hυ_h‖ and |ζ₃|/‖grad_hυ_h‖ (Fig. 10c and Figs. 19a,b respectively) show that W_D > 1 in the water parcels ofthe AJ (and not in the WAG) before arriving to the impinging region. However, when the AJ and the right flow branch are deflected by the coast, the rotational term becomes important except in a narrow area in the impinging region where W_D < 1. The above results provide the spatial scale in the impinging region where the rotational term is negligible in the acceleration field, and therefore provide the spatial scale where irrotational streamfunctions like ψ may be applied.

In order to compare the impinging process in the Alboran Sea with the analytical results of Whitehead (1985b) it is required to assign an asymptotic angle of incidence to the impinging current. Because the impinging current has streamline curvature, there is no objective way for providing such an angle of incidence. Assuming that this angle is taken as zero, that is, the impinging jet is consider to be normal to the wall, Whitehead’s results predict that the percent volume of fluid deflected to the right is 65% and 75% for the zero-and constant-potential-vorticity jets, respectively. The geostrophic transport function computed from the same data we have analyzed (Viúdez et al. 1996a, Fig. 13) shows that the deflected geostrophic transport is ∼2 Sv (Sv ≡ 1 × 10⁶ m³ s⁻¹) to the right (WAG), and ∼1 Sv to the left (AJ). This corresponds to 66% of volume deflected to the right, a result that is consistent with Whitehead’s theory in spite of the different jet configurations and different wall geometry (vertical vs sloping coastal boundary).

On the other hand, Whitehead and Miller (1979) and Whitehead (1985a), based on their laboratory simulation of the WAG, pointed out that the growth of the gyre may be due to the position of the stagnation point of the jet relative to coastal features. In their simulations the growing of the gyre was stopped by the basin geometry. However, since the WAG is rarely found to occupy the entire upper layer of the western Alboran basin [we know of very few cases when this happened (Lanoix 1974; Cano 1978)], it appears that, for some reason, these laboratory experiments exaggerated the effect of the impinging process on the WAG circulation. It should also be noted that in the laboratory experiments with approximate coastal geometry (Whitehead and Miller 1979) it was required to smooth Cape Tres Forcas (the cape located on the African coast at 3°W) from the coastal geometry in order to develop a gyre.

The results concerning the impinging process, based on CTD data, show that, since the stagnation surface is tilted in the vertical, the magnitude of the streamwise vorticity increases due to the vertical shear in the across stream direction with the upper layer flowing to the left (facing downstream) and the lower layer flowing to the right. The impinging process described here is therefore a 3D feature. We, however, consider that no further conclusions concerning the remote origin of the water on both sides of the stagnation surface can be made from the CTD results presented here. Two main effects prevent us from this, namely, (i) the discrete CTD sampling and (ii) the diabatic and frictional effects that prevent surfaces of constant potential density and surfaces of constant potential vorticity from being material surfaces.

We would like to emphasize that the results presented here have been concerned with the so-called external flow and not with the turbulent flow in the viscous region. Although we have borrowed this flow division from the more general impinging process in fluid mechanics without proving it totally (because of the lack of measurements over the continental shelf), some results presented here (especially those concerning the irrotational characteristics of the impinging flow) point out that such a division may be acceptable in the oceanographic context. As a further consequence, since the impinging current has streamwise vorticity, it would be interesting to verify experimentally if vorticity amplification and an increase of turbulence occur over the African continental shelf, as it occurs in quasi two-dimensional impinging (nongeophysical) flows. Another possible phenomenon associated with an impinging process that may deserve further study in the oceanic case is the existence of hydraulic jumps, or standing waves, (e.g., Watson 1964; Higuera 1994; Fedorov and Melville 1996). If such a hydraulic jump exists, it could be observed along the African coast.

Impinging jet processes have received little attention in physical oceanography. For example, the study about the opposite phenomenon, that is, coastal current separation, is much more developed (e.g., Ou and De Ruijter 1986; Page 1987; Signell and Geyer 1991; Klinger 1994). We think that this is because the impingment process, although it may occur frequently in the ocean, is not usually a steady process (the temporal evolution of an eddy forced against a boundary has been studied by Shi and Nof 1993). Our main objective has been therefore to study, in a broad way, the main kinematic characteristics of an impinging jet in the Alboran Sea, and to relate this process with other disciplines of fluid mechanics. Since the presence the Atlantic jet is assured by the location of the Strait of Gibraltar, the impingment process in the Alboran Sea case is, concerning the steadiness, a privileged one.

4. Summary

We have described an oceanic impinging jet process in terms of its density, curvatures, backing, vorticity, and deformation characteristics. In the impingment region the acceleration field is divergent, which is related to the fact that the magnitude of deformation is larger than the magnitude of rotation. It also has been found that the stagnant streamsurface does not lie in a vertical plane but tilts in the opposite direction to the vertical tilting of isopycnals. The flow upstream of the stagnation point is characterized by backing, speed convergence, diffluence, and negative streamwise vorticity. The flow past the stagnation point is characterized by veering, speed divergence, confluence, and positive streamwise vorticity. The stretching axes of a fluid element approaching the coast experience a rotation relative to the flow direction in such a way that they become almost parallel and normal to the coast in the impinging region. Only in a narrow area of that impinging region can the current be considered irrotational.