Modelling processes influencing wind-induced internal wave generation and propagation

ARTICLE IN PRESS Continental Shelf Research 24 (2004) 2245–2271 www.elsevier.com/locate/csr Modelling processes inﬂuencing wind-induced internal wav...

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ARTICLE IN PRESS

Continental Shelf Research 24 (2004) 2245–2271 www.elsevier.com/locate/csr

Modelling processes inﬂuencing wind-induced internal wave generation and propagation Alan M. Davies, Jiuxing Xing Bidston Observatory, Proudman Oceanographic Laboratory, Birkenhead, CH43 7RA, Merseyside, UK Available online 19 October 2004

Abstract The importance of the oceanic lateral boundary layer mixing in the shelf edge region upon oceanic mixing and hence ocean circulation and climate is brieﬂy reviewed with references to the literature for detail. The role of internal waves in this mixing and the various processes producing these waves is considered. References to the literature and numerical calculations using a three-dimensional model are used to demonstrate the importance of internal waves in mixing. Comparison of results from idealized cross-sectional and single point models illustrate some limitations of single point models in reproducing the mixed layer thickness and associated current proﬁles. In nearshore areas where the water is well mixed, calculations show that internal wave generation occurs at the front between the stratiﬁed and coastal region. Non-linear effects due to vorticity in the frontal jet, and sloping isotherms can lead to internal wave trapping with locally enhanced turbulence and mixing. The importance of using a three-dimensional model that can take account of along and across shelf internal pressure gradients in generating internal waves is demonstrated. The non-linear coupling between near-inertial motion and the internal tide in shelf edge regions is shown to be an important process in oceanic lateral boundary mixing. The need for future high resolution models that can take account of internal waves and associated mixing, together with accurate and comprehensive data sets for validation, is stressed in the ﬁnal section. r 2004 Elsevier Ltd. All rights reserved. Keywords: Numerical model; Internal waves; Mixing; Internal tide; Review

1. Introduction With increasing interest in the role of oceanic models in predicting climate change, it has become important to examine the role of small scale Corresponding author. Tel.:+151-653-8633; fax:+151-653-

6269 E-mail address: [email protected] (A.M. Davies).

mixing in these models and how it affects the large scale circulation. Recent ideas (Munk and Wunsch, 1998) suggest that lateral boundary layer mixing in the ocean and its diffusion into the interior has a major effect on the ocean circulation. Numerical calculations using ocean circulation models with enhanced mixing in shelf edge regions (Samelson, 1998; Spall, 2001) have shown that this mixing has a signiﬁcant inﬂuence upon

0278-4343/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.csr.2004.07.006

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the large scale circulation. Two major sources of energy for mixing are the tides and wind. The production of internal tides at the shelf edge (e.g. Xing and Davies, 1998) and their subsequent propagation into the deep ocean is one process that can inﬂuence oceanic mixing. A second major source of mixing is the generation of inertial oscillations and near-inertial internal waves (Garrett, 2001) by the wind. Since these are at the resonance frequency (namely the inertial frequency), appreciable currents can be generated by light winds. In near coastal and shelf edge regions near-inertial currents are accompanied by a 1801 phase shift across the thermocline, and an associated shear production of turbulence. The shelf edge region in particular is an important area for mixing which with associated small-scale features (Davies et al., 2003) determining shelf–ocean exchange (Davies and Xing, 2001). Consequently enhanced mixing in this region inﬂuences the large scale oceanic circulation. In this paper we will initially give a brief review of the processes that give rise to internal wave production. In particular the role of the wind and coastal fronts. The inﬂuence of non-linear effects will be stressed. Subsequently we will review various means that have been used to account for internal wave mixing in point models. In the main bosdy of the paper, following the development of a three dimensional model, the limitations of using one- and two-dimensional models compared with a full three-dimensional model of windinduced effects in stratiﬁed sea regions will be examined. The non-linear-interaction between inertial oscillations and the internal tide will be considered in the later part of the paper.

2. Overview of previous internal wave modelling and associated mixing processes Internal wave generation in the ocean surface layer in response to Ekman suction associated with horizontal changes in wind stress due to travelling storms has been studied by a number of authors (e.g. Gill, 1984; Greatbatch, 1983) but is outside the scope of the present paper where we concentrate upon the coastal ocean response to a spatially

uniform wind ﬁeld. In the coastal ocean the presence of the coastline, and the associated upwelling/ downwelling gives rise to internal wave generation even when the wind stress is uniform. In the absence of a coastline, assuming a homogeneous water column, and constant eddy viscosity, gives the classic Ekman problem, in which a suddenly applied wind ﬁeld gives rise to inertial oscillations and a wind drift current. As time progresses the wind’s momentum diffuses to depth, and inertial energy in the surface layer decreases (Davies, 1985a) with bottom friction leading to rapid damping in a shallow sea region (Davies, 1985b). If the water column is stratiﬁed, then vertical eddy viscosity is reduced at the level of the thermocline, and there are persistent inertial oscillations in this region (Davies, 1985a) as found in oceanic observations (Weller, 1982). In a linear model which forms the basis for mixed layer dynamics a horizontally spatial distribution of these oscillations would be expected. However, as shown by Weller (1982), convergences and divergences in the background ﬂow lead to changes in amplitude, phase and frequency of the nearinertial response. This produces an Ekman pumping effect and allows inertial energy to escape below the mixed layer. In the presence of a coastline a gradient current is produced which modiﬁes the Ekman drift current, and gives rise to a ﬂow at depth (Davies, 1985b). Under stratiﬁed conditions besides a surface elevation gradient, internal pressure gradients arise and internal waves are produced. Early calculations with cross-sectional models using either a layered (Millot and Crepon, 1981), functional, with a rigid lid (Kundu et al., 1983) or ﬁnite difference representation in the vertical (Tintore et al., 1995), showed that the initial response to wind forcing was the generation of inertial energy in the surface layer. At the coast this energy spread to depth (Kundu, 1984; Kundu et al., 1983) and slowly moved offshore. In order to set the present calculations in the broader context of this early work, we will review some of this previous research in more detail. The two layer model of Millot and Crepon (1981) showed that the currents associated with the inertial oscillations were clockwise polarized, with

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a 1801 phase shift between the layers and the ratio of current magnitudes between layers depending upon layer thickness. The 1801 phase shift in inertial oscillations between the surface and bottom layers was also found in an analytical model of a channel (Krauss, 1979). Based on measurements taken in the Adriatic Sea, Orlic (1987) reached similar conclusions to those of Millot and Crepon (1981). Inertial currents were accompanied by thermocline displacements, with the amplitude of the currents decreasing as the coast was approached. Based upon the linear dynamics of two layer ﬂow in a rectangular channel, and ignoring barotropic effects, Orlic (1987) showed that a ﬁrst mode baroclinic standing internal wave could explain the major features of the observed near-inertial currents in the region. The decay of near-inertial currents was shown to be due to bottom frictional effects, which had also been shown in shallow seas by Davies (1985b). As with all layered models a constant depth was assumed, and in all models discussed so far the source of internal wave generation was the vertical displacement of the density surfaces intersecting the coast. More recent calculations (Chen and Xie, 1997) were concerned with offshore variations in internal wave energy in near coastal regions including the shelf edge, and showed that the offshore distribution of near-inertial energy was determined by the internal stress and water depth. These authors used a cross-sectional version (see Chen and Beardsley, 1995) of the Blumberg and Mellor (1987) model to simulate the across-shelf distribution of inertial energy measured during the Texas–Louisiana Shelf Circulation Study. They also used a linearized analytical model to understand the major results from the more complex model. This model showed that within a linear approximation, and neglecting near-inertial internal wave propagation, the cross–shelf variation of near-inertial oscillations was controlled by the cross shelf gradient of surface elevation and vertical gradient of the Reynolds stress. An extension to the analytical work of Chen and Xie (1997) was performed by Lewis (2001) who examined the across shelf variation in amplitude of near-inertial current oscillations. An analytical

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model based on a vertically integrated barotropic across shelf slice model, showed that the offshore extent of the region of increasing magnitude of inertial oscillations (from zero at the coast) was of the order of 60 km. This was consistent with an exponential decrease of sea surface elevation gradient based on a barotropic radius of deformation. The coastally induced sea surface elevation gradient was due to the on/offshore currents associated with the inertial oscillations. The model study showed that the offshore increase in amplitude of near-inertial oscillations was not due to any frictional or non-linear effect but was related to spatial variations in the amplitude of free surface elevation gradients. This inﬂuence of the sea surface elevation gradient had not been taken into account in the early work of Kundu et al. (1983) and Kundu (1984) where a rigid lid had been assumed. Measurements using both HF Radar and ADCPs taken in the coastal region of the Ebro Delta in the Mediterranean (Rippeth et al., 2002) also showed a near coastal reduction in nearinertial oscillations, with current proﬁles characterized by a 1801 phase shift across the thermocline as found by Millot and Crepon (1981) and Orlic (1987). In parallel with cross-sectional modelling, simple single point models have been used to explain the observed (Rippeth et al., 2002; Knight et al., 2002) 1801 phase change across the thermocline in near coastal regions. The simplest explanation is the necessity to satisfy the no normal ﬂow condition at the coast namely that the vertical integral of the current is zero (De Young and Tang, 1990). An alternative (Davies and Xing, 2003a, b) which we will show is successful in the case of periodic wind forcing is to assume a balance between sea surface elevation gradient and wind stress forcing. Namely a local gradient current which is added to the Ekman drift current. A similar approach was used by Rippeth et al. (2002). Results from such single point models could reproduce the 1801 phase shift, but of course, could not account for the spatial variability in amplitude of the near-inertial oscillations in the coastal region. As shown by Davies and Xing (2003a, b), Xing and Davies (2003) (see also

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results in this paper), a single point model has very limited applicability. In particular, the primary problem with single point models, even ones in which the decay of inertial oscillations in the surface layer is reproduced by the addition of a linear friction terms, and mixing formulation has been adjusted (Mellor, 2001), is that they cannot take account of energy loss to the surrounding region by internal wave radiation. This loss of energy has an effect upon the mixed layer depth. In addition mixing in a region due to internal wave propagation into the region is not included. Also trapping of inertial energy in regions of negative vorticity and enhanced vertical mixing associated with this cannot be accounted for in a single point model (see later). Although single point models have major limitations in stratiﬁed conditions these models have been used very successfully in homogeneous conditions. This use is brieﬂy examined here in order to highlight issues that are relevant to the stratiﬁed problem. Craig (1989) showed that under the assumption of a constant eddy viscosity, the depth of penetration of the wind’s momentum was inversely proportional to the differences between the wind’s frequency and the inertial frequency. Consequently at the inertial frequency the wind’s momentum could penetrate to the sea bed. However, under stratiﬁed conditions, stable stratiﬁcation at the level of the thermocline reduced the eddy viscosity to near zero, and hence the wind’s momentum was conﬁned to a slab like layer above the thermocline. Shear production of turbulence across the thermocline, could give rise to a non-zero eddy viscosity and diffusivity, and hence the surface mixed layer depth increased with time. The parameterization of mixing in a single point model has been the subject of a number of studies. As discussed previously, one difﬁculty with a single point model in stratiﬁed conditions, is that it cannot allow for internal waves to radiate energy out of the region. Also it does not account for advection into the domain or internal wave propagation into the area. A detailed comparison of a number of single point models using different turbulence closure parameterizations and measurements was made by Burchard et al., (2002).

Differences between the various turbulence closure schemes was found to be smaller than differences in dissipation rate found in the measurements due to distances between measurement sites, degree of undersampling, differences in turbulence probe characteristics, and instrument errors. All turbulence models used the same external forcing, and a nudging of the computed temperature towards observations had to be included. In some of the turbulence energy models the macro length scale L was limited by the buoyancy as formulated by Galperin et al. (1988), namely L2 pL2min ¼

0:56k N2

with k turbulence energy, and N buoyancy frequency. This together with setting a minimum kmin of order (1 106 J kg1) (Bolding et al., 2002) for the turbulence energy was to take account of internal wave effects. Including constraints on L and k, ensured that the background eddy viscosity in the region of the pycnocline was consistent with observations of Van Haren (2002). However, as we will show later, and as suggested by Bolding et al. (2002) this may be too simple a parameterization of internal wave mixing. In essence the choice of kmin can be regarded as a tuning coefﬁcient, designed to ensure the correct depth of the thermocline (Bolding et al., 2002), by adjustment of the mixing across the thermocline. The fact that a single point model cannot take account of vertical motion of the pycnocline on a short time scale, and this affects the heat capacity of the surface layer was stressed by Bolding et al. (2002). When the single point model was adjusted to allow for inﬂow/outﬂow of water above and below the thermocline, so that the vertical movement of the thermocline corresponded to that found in observations, this improved the temperature distribution, although a relaxation to the observed temperature ﬁeld was required in the upper 20 m of the water column. Such an adjustment to observations was required in order to obtain the observed turbulence ﬁeld. Luyten et al. (2002) also considered the problem of determining the correct level of mixing within the thermocline in a one-dimensional model and suggested a third stability coefﬁcient connected

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with the diffusion of turbulence energy. In this approach the vertical transport of turbulence energy becomes important in the case of strong stratiﬁcation. Consequently the choice of an appropriate stability coefﬁcient for turbulence energy diffusion then becomes crucial. Comparisons of this approach with the use of limiting conditions on turbulence energy, mixing length or dissipation rate, suggested that the use of limiting conditions was most effective. Luyten et al. (2002), also points out that the magnitude of surface inertial oscillations in a single point model are signiﬁcantly over-estimated, with solutions in the literature (e.g. Mellor, 2001, and references therein) reducing their magnitude by adding a damping term in the momentum equations. Although this may be appropriate at a deep water site, Luyten et al. (2002) point out that in shallow water this would add an artiﬁcial damping in the tidal bottom layer. Since in a region of signiﬁcant tidal currents, U* at the bed is dominated by these currents, changes in U* would inﬂuence the near bed value of dissipation e which is given by ¼ U 3 =K ðzh þ z0 Þ; where zh is height above the bed, K Von Karman coefﬁcient and z0 bottom roughness. The models discussed above are primarily based upon the linear form of the hydrodynamic equations. However, non-linear effects due to vorticity associated with background ﬂows have an important inﬂuence upon the propagation of near-inertial internal waves and hence mixing. The interaction of near-inertial internal waves and background vorticity (Z) due to a steady ﬂow such as that associated with a frontal jet, was examined by Mooers (1975) and Kunze (1985) using an analytical model and ray tracking methods. The presence of this ﬂow changed the background vorticity from f, to an effective value f eff ¼ f þ Z=2: Waves which are generated or forced at a frequency oi are free to propagate both horizontally and vertically in the region where wi 4f eff : However their propagation can be blocked by a region in which f eff 4oi leading to a build up of near-inertial internal wave energy and hence increased mixing. In the case of sloping density surface which occur in the region of a front, then interaction can occur between the propagating internal wave and the sloping density front. In this

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case the wave frequency is given in a crosssectional model by sl ¼ f eff S2 N 2 ; (where S ¼ M 2 =N 2 is the slope of the iso-pycnal surface with N 2 ¼ g=r @ r=@z and M 2 ¼ g=r @r=@z is the equivalent in the horizontal). Thus inertial internal waves with frequencies between the anomalously low-frequency sl and the effective frequency feff can be trapped on the positive vorticity side of a front in regions of sloping density surfaces. These effects are illustrated in this paper in terms of a surface front. The associated differences in mixing between trapped and propagating near-inertial internal waves are also discussed. Changes in the local inertial frequency due to horizontal shear associated with alongshelf ﬂows was shown by Federiuk and Allen (1996) to increase near-inertial internal wave energy on the inner shelf during upwelling conditions in a fully non-linear model. However, when the model was linearized by removing the non-linear momentum terms this intensiﬁcation was no longer present. Lerczak et al. (2001) reported modulations of the along shelf ﬂow, that produced a change in the effective local Coriolis parameter by as much as 50%. At a latitude of 301N where the diurnal sea breeze frequency is close to the local inertial frequency, these modulations of feff could give rise to energetic sea breeze forced baroclinic motions, modulated by changes in vorticity of the alongshelf ﬂow. Although these effects involve the non-linear momentum terms, they have been examined (e.g. Mooers, 1975; Kunze, 1985) by assuming a steady background sheared ﬂow and hence wave–wave interaction (Muller et al., 1986) in which energy cascades from low frequency to high-frequency waves is excluded. However, as we will show energy transfer through the non-linear momentum terms, in a similar way in which energy is transferred from the principal lunar semi-diurnal tide (the M2 tide) to its higher harmonics (denoted M4, M6, etc., Kwong et al., 1997), is important in coupling together M2 internal tides, and wind energy at the inertial frequency f, to give energy at the sum of their frequencies (denoted fM2). This is important as a means of enhancing mixing through internal waves of tidal and wind origin. This process is the ﬁrst step in the cascade of energy to high frequencies through the non-linear

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terms. Both on and off-shelf measurements have shown that there is signiﬁcant energy in both shallow seas (van Haren et al., 1999; Gemmrich and van Haren, 2002) and oceanic currents (Mihaly et al., 1998) at the fM2 frequency to indicate that there is signiﬁcant coupling between the wind and the tide. In subsequent sections of this paper we will use numerical model results to extend and elaborate the overview given here. The limitations of oneand two-dimensional (cross-sectional) models compared to a full three-dimensional model will also be discussed. The form of the three-dimensional numerical model, the associated crosssectional and point models together with the turbulence closure model used to investigate the generation and propagation of near-inertial internal waves is given in the next section. In subsequent sections, cross-sectional models with idealized topography are used to examine the limitations of single point models. In a later section a three-dimensional model with realistic topography representing the Balearic Sea is used by comparison with a cross-sectional model to show the importance of along shelf ﬂow and internal pressure gradients in determining the internal wave ﬁeld. The inﬂuence of vorticity associated with along frontal jets upon the trapping, propagation and mixing due to internal waves is considered in the latter part of the paper. Other non-linear effects due to the momentum advection terms in the hydrodynamic model are illustrated using the coupling of the M2 internal tide and inertial oscillations. The ﬁnal part of the paper draws some conclusions as to the processes generating internal waves and inﬂuencing their propagation. Some conclusions as to their importance in mixing and inﬂuence on large scale models and suggestions for further work is given in the ﬁnal section.

3. The hydrodynamic model equations 3.1. The three-dimensional model A three-dimensional, free surface, baroclinic model (Xing and Davies, 2001), with vertical eddy

viscosity and diffusivity parameterized using a turbulence energy model (Xing and Davies, 2001), is used in the calculations. A brief overview of the model is presented here. The model equations in s coordinates, where s ¼ ðz BÞ=H (yielding s ¼ 0 at the surface and s ¼ 1 at the seabed), with the conventional Boussinesq and hydrostatic approximations, are given by * @Hu @Huo þ r ðV HuÞ þ fHv @t @s @B ¼ gH þ BPF x @x 1 @ @Hu Þ þ F u; þ 2 ðK m @s H @s * @Hv @Hvo þ r ðV HvÞ þ þ fHu @t @s @B ¼ gH þ BPF y @y 1 @ @Hv Km þ 2 þ F v; @s H @s

@B þr @t

Z

0

ð1Þ

ð2Þ

~ ds HV

¼ 0;

(3)

1

@HðT; SÞ ~HðT; SÞÞ þ @HoðT; SÞ þ r ðV @t @s 1 @ @HðT; SÞ Þ þ F ðT;SÞ ; ¼ 2 ðK h @s H @s

ð4Þ

@P ¼ rgH; @s

(5)

r ¼ 1000:0 þ st

(6)

with st ¼ 28:152 0:75T 0:00469T 2 þ ð0:802 0:002TÞðS 35Þ:

ð7Þ

The vertical velocity o was computed from a diagnostic form of the continuity equation. In ! these equations, V ¼ ðu; v; oÞ are the velocity componets corresponding to the (x, y, s) coordinates with x the offshore and y the alongshore coordinate; r is the horizontal divergence operator; r is the density; T is the temperature; S is the salinity; H is the total water depth; B is the

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sea surface elevation above the undisturbed level; z is the water depth increasing vertically upwards with z ¼ B at the free surface and z ¼ h at the seabed; f is the Coriolis parameter; g is the gravitation acceleration; t is time; Km and Kh are the vertical eddy viscosity and diffusivity coefﬁcients; Fu, Fv and F(T,S) are the horizontal diffusion terms; P is the pressure ﬁeld; BPFx and BPFy are the baroclinic pressure gradient force terms which are given in Xing and Davies (2001). They are computed in such a way that errors due to vertical coordinates transformation can be reduced (Xing et al., 1999). The horizontal diffusions have a biharmonic form on the sigma surface namely, F ðuvÞ ¼ HBm r4s ðu; vÞ; F ðTSÞ ¼ HBh r4s T T ref ; S S ref ;

ð8Þ

where Bm and Bh are coefﬁcients of horizontal diffusion terms for momentum and density (temperature and salinity), respectively with Tref, Sref background mean values. In the present application the salinity ﬁeld is kept constant at S=35 psu. The vertical eddy viscosity (Km) and eddy diffusivity (Kh) are calculated using a two equation turbulence energy model which predicts turbulence intensity and mixing length, both of which depend upon the local Richardson number. Details of the model are given in Xing and Davies (2001) and will not be repeated here. For a detailed discussion of a range of turbulence energy models and their formulation the reader is referred to Blumberg and Mellor (1987); Luyten et al. (1996, 2002); Burchard et al. (2002); and Bolding et al., (2002). The normal component of ﬂow was set to zero at the land boundary and a radiation condition was used along open boundaries. A slip condition was applied at the sea bed and the surface stress was set equal to the wind stress. For temperature the heat ﬂux was zero at sea surface and sea bed. Further details of the model are given in Xing and Davies (2001). Discretization was accomplished using an Arakawa C grid in the horizontal with grid resolution 2 and 3 km across and along shelf. In the vertical a non-uniform grid spacing of 27 levels with enhanced resolution in the surface high shear region.

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3.2. Cross-sectional model In some calculations to compare results from a cross-sectional model, with those from a full threedimensional calculation, and hence appreciate the limitations of two-dimensional vertical sectional models, a number of cross-sections are examined. In these calculations the coast is at x ¼ 0 with the model extending up to 200 km offshore into deeper water as in the full three dimensional model. The fundamental difference between the cross-sectional and three dimensional model is that there is no along shore y-dependency and hence the terms @B=@y and BPFy which involve surface and internal pressure gradients in the y direction are zero. Also by assuming uniformity in the along shelf direction, the @v/@y term in the continuity equation is zero. Consequently although an along shelf ﬂow can be generated this is uniform in the along shelf direction and cannot produce a change in sea surface elevation. This is an important difference from a three dimensional model where along shelf variations in the ﬂow, and pressure gradients can arise. In the non-linear case terms such as v@u=@y and v@v=@y are also absent. 3.3. Point model In some calculations comparisons will be made with single point models. As demonstrated by Ekman, under conditions in which there is no appreciable internal pressure gradient, the velocity at a point can be separated into that due to wind forcing (the ‘‘wind drift’’ current) and sea surface elevation gradients (the ‘‘gradient’’ current). Thus, for a generalised external barotropic forcing having components Fx, Fy the single point equations are given by, @u 1 @ @u fv ¼ F x þ 2 AV ; (9) @t @s H @s @v 1 @ @v þ fu ¼ F y þ 2 AV : (10) @t @s H @s In the absence of bottom stress, and under steady state conditions, with near zero geostropic ﬂow Fx, Fy can be given by F x ¼ tsx =rH; F y ¼ tsy =rH; (where tsx ; tsy are components of surface wind

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stress) in essence sea surface elevation gradient is balanced by local wind stress. In the calculations described later, the validity of this approximation is examined, as are the situations under which a single point model can given a current proﬁle comparable to that found in a three-dimensional model.

4. Idealized cross section calculations In an initial series of calculations designed to examine the coastal generation of waves, and the extent to which a point model can reproduce current proﬁles away from the coastal boundary, the water depth was ﬁxed at h=100 m. Eddy viscosity AV and eddy diffusivity KV were ﬁxed at 0.001 and 0.0001 m2 s1 in both a cross-sectional model with a coast on the west side, and a single point model. The non-linear momentum advection terms were removed, although the role of these will be discussed in later calculations. An initial temperature ﬁeld with a surface value of 16 1C and a thermocline at about 40 m was used in both calculations. Wind forced motion due to a sinusoidal onshore/offshore wind forcing at a frequency of taken at the inertial, sub-inertial and super-inertial frequency was examined using both the cross-sectional and single point model.

(a)

4.1. Forcing at of ¼ f In the cross-sectional model under homogeneous conditions the sea surface elevation at the coast and off-shore rapidly responds to sinusoidal variations in the wind stress with elevation increasing/decreasing in response to on-shore/ off-shore winds. The surface elevation gradient term F x ¼ g@z=@x was accurately approximated by tx/rh when h=100 m, although when h was reduced, bottom friction became important and this balance no longer occurred. In the stratiﬁed case besides changes in sea surface elevation, internal waves are produced at the coast and slowly propagate offshore. Time and distance plots of Fx and the depth mean pressure gradient BPF x in the coastal region (Fig. 1) show that the internal pressure gradient extends about 20 km

(b) Fig. 1. Time variation of contours of (a) F x ¼ gð@z=@xÞ and (b) depth mean internal pressure gradient ðBPF x Þ in the nearshore (50 km from the coast) region due to a sinusoidal wind forcing at the inertial frequency of ¼ f :

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(i)

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(ii)

Fig. 2. Time series of temperature contours, u and v components of velocity, together with vertical velocity at (i) Posn 1, (ii) Posn 2 (but not vertical velocity).

off-shore. Outside this region which we will deﬁne as the coastal boundary layer the internal pressure gradient is negligible and the surface elevation gradient is in close agreement with that found under homogeneous conditions and given by tx =rh: Time series of the temperature at a point 6 km from the coast, and contours of the u and v components of velocity show (Fig. 2(i)) oscillations at the forcing period, reaching a periodic steady state after 8 days. Although the model continues to be forced by a periodic wind stress at the inertial period, by this time an along shore ﬂow has developed, leading to an increase in surface elevation and internal pressure gradients at the coast and the radiation of internal waves. These processes remove wind energy from the region and prevent the continued growth of inertial oscilla-

tions that would occur in a single point model. The current proﬁles (Fig. 2(i)) exhibit a 1801 change in phase at about the depth of the thermocline. Time series at a point located at the edge of the coastal boundary layer, show (Fig. 2(ii)) a vertical diffusion of the temperature ﬁeld, although there is no signiﬁcant internal wave activity. Surface u and v components of velocity show inertial oscillations, phase shifted by 1801 from those at depth (Fig. 2(ii)) which are stronger than the currents at 6 km from the coast. In essence at positions outside the coastal boundary layer, the inﬂuence of the internal pressure gradient is negligible. In deep water (h greater than 100 m) the effects of bottom friction are small, and current proﬁles are determined by the wind stress and local sea surface elevation gradient. As shown by Davies and Xing (2003a, b) at these locations, currents due to a

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horizontally uniform sinusoidal wind stress forcing can be accurately reproduced in a single point model forced by the surface wind stress, with the elevation gradient terms approximated by (tx, ty)/ rh. Physically the wind stress forcing produces ﬂow in the surface layer with the elevation gradient term acting as a body force to produce a uniform ﬂow through the water column phased shifted by 1801 from the surface current. As we will show in the next section, the extent to which this single point model can be applied depends upon the thickness of the coastal boundary layer which is a function of forcing frequency of. 4.2. Forcing at super-inertial and sub-inertial frequencies To illustrate the extent to which near-inertial internal waves propagate away from the coast, the cross-sectional model was forced at both superinertial frequencies of ¼ 1:23f and sub-inertial of ¼ 0:77f : Time and space contours of g@z=@x and depth mean internal pressure gradient BPX computed with of ¼ 1:23f show (Fig. 3a) that the coastal boundary layer has a signiﬁcantly larger off shore scale than for of ¼ f : An instantaneous picture of the cross-shore distribution of temperature, u, v and w (Fig. 3b) shows that there is signiﬁcant displacement of the temperature surfaces due to internal wave activity at up to 40 km from the coast. However the magnitude of the vertical velocity producing this displacement, decreases with distance away from the coast. Contours of u and v components of velocity, and a detailed harmonic analysis (not presented) show regions of enhanced surface current amplitude at the forcing frequency in the coastal boundary layer, with currents at depth phase shifted by 1801 from the surface current. The reason for these regions of enhanced magnitude will be discussed later. As in the case of of ¼ f ; outside the coastal boundary layer, current proﬁles could be reproduced by the single point model. In the case of forcing at the sub-inertial frequency of ¼ 0:77f (the sea breeze period), the off-shore extent of the coastal boundary layer (not shown) was substantially reduced below that

found with of ¼ f : The case of the sub-inertial forcing (Davies and Xing, 2003a, b) showed that the single point model was accurate provided it was not applied in the near coastal (less than 10 km) region. These results explain why single point models such as those used by Rippeth et al. (2001), Knight et al. (2002) and De Young and Tang (1990) are successful when applied outside the coastal boundary layer. In essence these models only account for wind forcing, and the presence of the coast in terms of Fx related to the local wind, or by ensuring that the mean ﬂow is zero. However as we have shown this is sufﬁcient to reproduce the observed 1801 phase shift in currents across the thermocline, without taking account of the internal wave ﬁeld. In the next section we will show that this is not sufﬁcient to reproduce some of the features of the response to wind induced inertial currents. 4.3. Response to a wind impulse In reality a sudden increase and decrease in wind magnitude occurs rather than sinusoidal variations at a given frequency. This effect was examined by considering a wind pulse of 24 h duration during which time it changed from initially on-shore; reached a maximum of 0.5 Pa with an alongshore direction at 12 h, with magnitude decreasing to zero, and orientation changing from alongshore to offshore over the next 12 h. To illustrate the importance of current shear in determining the level of mixing, calculations were performed with the turbulence energy model. Time and space contours of sea surface elevation and internal pressure gradient (not shown) computed with the cross-sectional model exhibited inertial oscillations within a coastal boundary layer of the order of 20 km, comparable to that found with of ¼ f : An analysis of results (not presented) after an initial 4-day transition period showed that the response was mainly at the resonant period, namely the inertial period. An alongshore current was generated within the ﬁrst 12 h in response to the wind forcing which was conﬁned to the nearshore region and balanced by internal and surface pressure gradients.

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Fig. 3. (a) Time variation of contours of F x and depth mean internal pressure gradient in the nearshore region (50 km from the coast) due to forcing at of ¼ 1:23f computed with ﬁxed Av and K v values. (b) A ‘‘snap shot’’ at t ¼ 10 days of contours of temperature T1C, v(cm s1), u(cm s1) and w(cm s1) velocity through depth in the nearshore region due to forcing at of ¼ 1:23f computed with ﬁxed Av and K v values.

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Fig. 4. (a) Time series of temperature contours, u, and v components of velocity, at Posn 2. (b) Time series of temperature contours, u, v velocity computed with the single point model with F x and wind stress forcing using the turbulence closure scheme.

Previous calculations with ﬁxed viscosity and diffusivity showed that provided a location was chosen outside the coastal boundary layer, the current proﬁle computed with the single point model was comparable to that determined with the cross-sectional model. To examine if that is true in the present case, the same location 24 km from the coast was used. Time series of the temperature ﬁeld (Fig. 4a) is characterized by signiﬁcant mixing over the top 40 m during the ﬁrst day when wind forcing was present and surface eddy viscosity was large. As the wind stress falls to zero this viscosity rapidly decreases, although bottom frictional effects remain due to currents at depth, and this damps currents below the thermocline. Since this point is well removed from the coast, the effect of internal waves was small. The initial response of the temperature ﬁeld computed with the single point model (Fig. 4b) was comparable to that found in the crosssectional calculation. During the period of wind stress forcing the elevation gradient was non-zero and this forced currents at depth. However when the wind stress forcing drops to zero, this term

vanishes, and bottom frictional effects rapidly damp the current at depth (Fig. 4b), leaving inertial oscillations above the thermocline. The presence of stable stratiﬁcation associated with the thermocline prevents the wind’s momentum diffusing out of the surface layer, and these oscillations continue with little damping. This suggests that relating the local elevation gradient in a single point model to local wind stress may have limitations in the case of rapidly changing rather than sinusoidal winds. This problem together with the inability of single point models to loose energy by internal wave radiation explains in part differences between single point models and full three dimensional calculations. These differences are examined later in terms of a model of internal motion in the Balearic Sea region.

5. Inﬂuence of a sloping bottom and mixing parameterization In practice the coastal boundary is never vertical, and in this series of calculations a

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Fig. 5. (a) The amplitude (cm s1) of the u component of velocity at of ¼ 1:2f computed with the prognostic model with AV =K V ﬁxed. (b) As (a) but at of ¼ f : (c) As (a) but with a diagnostic density ﬁeld. (d) As (c), but for of ¼ f :

nearshore sloping topography with water depths increasing from 100 to 300 m over 40 km (a slope of 0.05%) was assumed. A similar situation would occur at a shelf edge where stratiﬁcation intersects a slope. In an initial series of calculations AV and KV were constant as previously. Since wind forcing at the super-inertial period gives rise to a signiﬁcant offshore boundary layer, calculations were performed with of ¼ 1:2f : Although forcing was at of ¼ 1:2f ; the sudden imposition of forcing excited energy at the inertial period. Consequently to examine the spatial distribution of the amplitude and phase of currents at both these periods, a harmonic analysis was performed. The spatial distribution of the u current amplitude at of ¼ 1:2f frequency (Fig. 5a) has regions of enhanced magnitude in the surface layer associated with internal wave propagation from the nearshore generation point. Calculations (not presented) showed that these were sensitive to both stratiﬁcation and bottom slope, in an analogous way to the offshore propagation of internal tides (Xing and Davies, 1998). The distribution of the u current amplitude at the of ¼ f frequency shows (Fig. 5b) a surface maximum at approximately 70 km offshore. To conﬁrm that these maxima resulted from internal wave propagation, the calculation was repeated with a diagnostic density ﬁeld (Xing and Davies, 2003). In this case a uniform offshore distribution in amplitude at both

the of ¼ 1:2f (Fig. 5c) and of ¼ f (Fig. 5d) was found. To determine the effect of changing the formulation of AV/KV (hence vertical mixing) upon the solution, the previous calculation with the prognostic density ﬁeld was repeated using the turbulence energy model. As previously the amplitude of the u component of current at of ¼ 1:2f has regions of enhanced amplitude comparable to those found previously (compare Figs. 6 and 5a). In the present calculation the surface value of AV is larger than previously, allowing more of the wind’s momentum to diffuse out of the surface layer, giving reduced amplitudes at the surface. In the next section using idealized models of internal wave generation in frontal regions we will show how non-linear effects associated with the momentum advection term can inﬂuence the propagation of near-inertial internal waves.

6. Frontal inﬂuence upon internal wave generation and propagation In the previous series of calculations it was assumed that the stratiﬁcation extended to the coast. Consequently near-inertial internal waves were generated at the coast. However in many cases the water shallows as the coast is approached and this together with strong tidal currents leads to

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Fig. 6. As Fig. 5a but with AV =K V from a turbulence energy model.

a well-mixed coastal region with stratiﬁcation offshore. For example the northern North Sea, where signiﬁcant inertial energy is found in the deeper water (Knight et al., 2002), although the coastal region is well mixed. In many cases the nearshore region is separated by a surface front from the offshore stratiﬁed water. To examine the inﬂuence of a surface front upon the generation of near-inertial internal waves in coastal regions, a series of idealized calculations were performed. Initially these were with an isolated surface front in an inﬁnite ocean, although subsequently the effect of adding a coastal boundary, was considered.

Fig. 7. Across shore distribution of initial temperature (1C) (thick solid line) and along shore velocity v (m s1) (contour interval 0.05 m s1, thin solid line, with zero contour dotted and dashed) in the frontal region.

6.1. Surface front in an infinite ocean In an initial calculation an inﬁnite domain with water depth h=100 m was used extending from west to east, with the front placed at x=25 km (Fig. 7). In subsequent calculations a coastal boundary is placed at x=0. In both cases the turbulence energy model was used to compute AV/KV. To the west of the front the water is well mixed at 13.5 1C, while to the east temperature varies from 16 1C (surface) to 13.51 (bed). In the frontal region there is an along frontal jet v, with a surface maximum of 35 cm s1 which is decreased over 15 km to 5 cm s1 (Fig. 7). This produced regions of positive and negative vorticity Z on the western and eastern side of the front of order 0.4 104 s1. To complement calculations considered in the next section, the region was taken at the latitude of the Mediterranean for which the inertial period is 18.4 h (inertial frequency f=0.9485 104 s1). Motion was induced by a

wind pulse of 24 h duration and a maximum of 0.5 Pa, aligned so as to give a mean ﬂow in the direction of the surface jet and initially gave rise to inertial oscillations in the surface layer. As time progresses, inertial energy diffuses out of the surface layer, and its distribution is determined variance contours Vu ¼ by ðu huiÞ2 ; where u is the across shore component of velocity, and angled brackets indicate averaging over an inertial period. Although this is not as accurate as determining the amplitude of the inertial oscillations by harmonic analysis it is a reasonable measure of the energy at the inertial period (Federiuk and Allen, 1996). Vu contours after 2 days, show (Fig. 8a) that on the western well-mixed side of the front there is very little inertial energy. In this region the vertical eddy viscosity is large (Fig. 8b), and inertial energy rapidly diffuses from the surface layer to depth where it is damped by bottom friction (Davies,

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(b) Fig. 8. (a) Contours of across shore distribution of variance Vu (103 m2 s2) in the frontal region at days 2 and 6, computed with the turbulence energy sub-model, and an inﬁnite region. (b) Contours of across shore distribution of vertical eddy viscosity Av (cm2 s1) at day 2 and 6 in the frontal region, computed with the turbulence energy model and an inﬁnite domain.

1985b). On the eastern side of the front stable stratiﬁcation at the level of the thermocline gives a near zero value of eddy viscosity, and inertial energy is trapped in the surface layer in the form of wind induced inertial oscillations.

On the positive vorticity (western side of the jet) super-inertial internal waves are generated at a frequency oi above inertial. Internal waves generated in this region for which oi lies in the range f eff ooi oN (with N the Brunt–Vaisalla frequency

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and f eff ¼ f þ Z=2 ), will propagate out of the region (Mooers, 1975; Kunze, 1985; Dale et al., 2001). Due to the well mixed nature of the water column on the western side of the front for which N=0, propagation is to the east. Internal wave generation on the negative vorticity (eastern) side of the front at a sub-inertial frequency cannot propagate horizontally, although vertical propagation is possible, giving a region of enhanced variance at depth (Fig. 8a). As time progresses, super-inertial internal waves propagate to the east giving rise to the surface maximum at x=62 km on day 6 (Fig. 8a) which propagates farther eastward to x=75 km by day 8. In the frontal region inertial energy propagates to depth along the sloping iso-pycnal surfaces giving a maximum at x=30 km at about 30 m below the surface, (day 6, Fig. 8a) which gradually spreads farther down the water column (day 8, not shown). A region of enhanced surface inertial energy appears to be trapped in the region next to the well-mixed water column. At this location surface eddy viscosity appears to be small (day 6, Fig. 8b) and the inertial energy only diffuses slowly out of the surface layer. It is evident that to the east of the front, although there is signiﬁcant inertial energy in the surface layer, there is none at depth. However measurements made in coastal regions (e.g. Knight et al., 2002) do show some inertial energy at depth. The role of the coastline in determining this will be discussed in the next section. 6.2. Surface front in proximity to a coastal boundary In this calculation the surface front was located 25 km to the east of a coastal boundary at x=0. After 2 days variance contours to the east of the front, in the region removed from the frontal inﬂuence, have a horizontally uniform distribution (Fig. 9), with inertial energy below the thermocline. This is due to the generation of inertial oscillations in the surface layer, and an associated coastal elevation response. This response drives inertial oscillations throughout the water column, phase shifted by 1801 from those in the surface layer, as discussed earlier in the paper. This is the

reason for the reduction in magnitude of Vu in the surface layer when coastal effects are included. As in the case of the front alone a region of surface intensiﬁed variance propagates to the east (see days 6 and 8 in Fig. 9), associated with internal wave propagation in this direction. Also there is evidence of a near mid-water maximum at x=40 km, and a surface maximum at x ¼ 20 km; analogous to those found in the front only calculation. This calculation clearly shows that the generation of signiﬁcant inertial energy at depth at offshore stratiﬁed locations is due to the coastal boundary. Also offshore propagating internal waves can occur even when the stratiﬁcation does not intersect the coastal and bottom topography. These internal waves originate from the region of the surface front.

7. Three dimensional calculations in the Balearic Sea Region Following a recent series of measurements in the Balearic Sea, situated in the northwest Mediterranean offshore of mainland of Spain (Fig. 10), a three-dimensional model of the region was developed to examine the effect of an along shelf ﬂow and wind forcing upon the Ebro plume (Xing and Davies, 2002). The model is used here to brieﬂy examine the inﬂuence of an along shelf ﬂow upon the distribution of near-inertial energy in the region. The model extends 400 km in the alongshore direction and 190 km offshore, with a uniform grid of 3 km alongshore and 2 km offshore. An irregularly spaced grid of 25 levels was used in the vertical, with grid reﬁnement in the surface layer. Other details of the model can be found in Xing and Davies (2002). The initial temperature distribution was characterized by a well-mixed 20 m thick surface layer at 16 1C. Below this was a thermocline of approximately 20 m extent within which the temperature diminished linearly to 14 1C. A subsequent gradual decrease to 13.4 1C at 100 m, and constant temperature below this was assumed. This temperature distribution was applied everywhere. Calculations were started from initial conditions of zero elevation and

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Fig. 9. Contours of across shore distribution of variance Vu (103 m2 s2) at days 2, 6 and 8 in the near coastal region computed with a coastal boundary and a front (situated 25 km off shore) using the turbulence energy sub-model.

motion. In the outputs presented subsequently, only a limited region of the model domain is shown and ﬂow at cross-section C3 (located at 180 km north on the wide shelf close to the Ebro mouth) is examined. In a more detailed study (Xing et al., 2004) ﬂow at cross-section C1 and time series at an offshore point (Posn A) on the shelf in 90 m water depth are also examined. 7.1. Spatial distribution of wind forced inertial energy Before considering the inﬂuence of an alongshelf ﬂow, the response to wind forcing alone is examined. Contours of variance V ¼0:5ðhðuhuiÞ2 þðvhviÞ2 iÞ in the surface layer (Fig. 11a), show a reduction in

variance in the coastal region due to the presence of the coastline, with bottom frictional effects also removing inertial energy in shallow water (Davies, 1985b). Contours of the across shelf variation of variance V u ¼ hðu2ðhuiÞ2 i along cross-section C3 show (Fig. 11b) an intensiﬁcation at about 45 km from the coastline. Comparison with a diagnostic model (not shown) in which internal waves were not present, showed that this intensiﬁcation was associated with the internal wave ﬁeld in a similar way to that found with the idealized cross-section model described previously. These internal waves are generated where density surfaces intersect the bottom topography in particular in the region between 10 to 20 km from the coast (Figs. 11c), and propagate away from the generation point

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Fig. 10. Topography of the Balearic Sea region, and location of the three-dimensional model, and cross sections C1, C3 and Position A.

following pathways that depend upon topography and stratiﬁcation. Although inertial energy in the surface layer decreases through mixing giving rise to a thicker well-mixed layer over which the inertial energy is spread, it is clear that a region of enhanced inertial energy at depth persists (Fig. 11b, day 10). 7.2. Spatial distribution of wind forced inertial energy, with a prescribed alongshelf flow to the south On the short time scale (of order 4 days) the spatial distribution of surface variance V computed with an alongshelf ﬂow (Fig. 12a) is comparable to that found without the ﬂow. A similar solution was found in a linear model (not shown) in which the non-linear momentum terms were omitted. However as time progresses the variance on the shelf and at the shelf edge decreases more rapidly than in the previous calculation. This decrease is clearly evident in the contours of V at day 10 (compare Figs. 12a and 11a), and across shelf contours of Vu (compare

Figs. 12b and 11b). As discussed previously in connection with the frontal model, when the nonlinear momentum terms are included, vorticity effects associated with the shear in the along shelf ﬂow can dampen the inertial motion and change its frequency (Weller 1982). These effects even in a diagnostic model in which internal waves are not present will change the distribution of /(u/ uS)2S and explain some of the differences between linear and non-linear solutions when an alongshelf ﬂow is present. In a prognostic calculation, besides this inﬂuence of the alongshelf ﬂow, the density ﬁeld also changes as shown in the across shelf temperature contours (Figs. 11c and 12c) due to the across shelf current induced by the along shelf ﬂow. This together with the possibility of advecting different water masses into a region inﬂuences the internal wave ﬁeld. Similarly in the presence of the alongshelf ﬂow there is increased mixing in the near bed region and a thicker bottom boundary layer than in the original calculation. The change in density gradient in the nearbed region produces differences in the generation and propagation of internal waves. Xing and Davies (1998) found the internal tide was inﬂuenced in a similar way by upwelling and downwelling favourable winds. In the non-linear model, the existence of regions of potential vorticity Z associated with shear in the alongshelf ﬂow, inﬂuences the propagation of near-inertial internal waves in an analogous manner to that found in the frontal calculations. However, for the along shelf ﬂow considered here, it appeared that the inﬂuence of ﬂow related vorticity upon the inertial oscillations was more important than its effect upon the internal wave ﬁeld. This together with its effect upon viscosity and bottom mixing appeared to have a greater inﬂuence upon the variance than differences in internal wave propagation. 7.3. Spatial distribution of wind-forced inertial energy with a specified along shelf flow to the north Changing the direction of the along shelf ﬂow besides modifying the distribution of Z, reverses the direction of across shelf ﬂow, from downwelling favourable, to upwelling favourable. Some

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(c) Fig. 11. (a) Contours of the variance V at the sea surface computed with the prognostic model at days 4 and 10. (b) Contours of variance hðu huiÞ2 i along cross-section C3, for days 4 and 10 computed with the prognostic model with wind forcing. (c) Contours of the time averaged (over an inertial period) temperature ﬁeld at day 10, over cross-section C3.

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(c) Fig. 12. (a) Contours of surface variance V computed with the non-linear prognostic model and an along shelf current to the south at days 4 and 10. (b) Contours of hðu huiÞ2 i along cross section C3, for days 4 and 10. (c) Temperature contours (1C) at cross section C3 for day 10.

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differences in surface variance (Fig. 13a) between this calculation and the previous one (Fig. 12a) are clearly evident, as are differences in the across shelf distribution of Vu (Figs. 13b, 12b and 11b). From Fig. 13b (day 4) it is clear that the location of surface and bed inertial energy maximum has moved onshelf, and surface inertial oscillations in the deep water are increased. This is due in part to the change in the temperature ﬁeld (Fig. 13c) and distribution of surface elevation (not shown). At the shelf edge the elevation gradient has decreased, due to a reduction in the along shelf ﬂow compared to previously. This change in along shelf ﬂow leads to a decrease in damping of inertial oscillations through both non-linear and viscous effects. The distribution of vorticity also changes. However as this is small it does not appear to have a signiﬁcant inﬂuence upon the propagation of near-inertial internal waves. The change in magnitude and direction of the alongshelf ﬂow leads to a reduction in the across shelf (u component) of velocity from an offshelf ﬂow of order 12 cm s1 (not shown) to an onshelf ﬂow of order 4 cm s1 (not shown), with an associated upwelling of temperature surfaces (Fig. 13c) compared with the downwelling and associated well-mixed bottom layer found previously (Fig. 12c). These changes inﬂuence the magnitude of inertial oscillations and the generation and propagation of near-inertial internal waves.

8. Non-linear interaction with the internal tide In the previous sections, the inﬂuence of the non-linear momentum terms was through vorticity associated with the presence of a background ﬂow. When the background ﬂow is steady the momentum terms can be linearized and analytical methods (Mooers, 1975; Kunze, 1985), show that the background ﬂow related vorticity inﬂuences the frequency of wind-induced near-inertial internal waves. However, the non-linear nature of the momentum advection terms means that through wave–wave interaction they can move energy to higher frequencies giving waves at a frequency which is the sum of the forcing frequency.

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Eventually these non-linear processes move energy from internal waves into turbulence (D’Asaro and Lien, 2002). In tides the non-linear interaction through the momentum terms in shallow water generates higher harmonics for example the MS4 tide from the lunar and solar semi-diurnal tides M2 and S2 (Kwong et al., 1997). Recent measurements in the North Sea (Van Haren et al., 1999) and in the deep ocean (Mihaly et al., 1998) have shown that coupling between the inertial energy at frequency f and the M2 internal tide can give rise to energy at the sum of these frequencies, deﬁned as the fM2 frequency. To try and understand the processes producing this, calculations have been performed with an idealized cross-sectional model of the region off the west coast of Scotland. The area was chosen because it is subject to signiﬁcant wind forcing and hence appreciable inertial oscillations exist, and has a strong M2 internal tide (Xing and Davies, 1997, 1998). The cross-sectional model starts from rest with horizontal temperature surfaces and barotropic tidal forcing at the M2 period introduced at the oceanic boundary using a radiation condition. After 5 days the internal tide was established and wind forcing in the form of a clockwise rotating pulse of 0.5 days duration and maximum magnitude of 0.5 Pa was applied everywhere. Following a spin up period, a 5 day time series was harmonically analysed to determine the amplitude and phase of the rotary components of current at the f, M2, M4 (the ﬁrst higher harmonic of M2) and fM2 frequency. Since the model, unlike measurements, is in essence ‘‘noise free’’ a 5 day time series was sufﬁcient to separate the various frequencies. Contours of the clockwise component at the inertial period, show (Fig. 14) surface amplitudes in the ocean exceeding 20 cm s1, with a decrease on the shelf and some inertial energy at depth as described previously due to the coastal effect. The internal tide was separated from the barotropic tide by subtracting a depth mean tidal current prior to the harmonic analysis. Away from the near bed region this gives the internal tide, however close to the bed frictional effects inﬂuence the barotropic tide and a spurious internal tide appears (Fig. 14). Contours of the amplitude of

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Fig. 14. Contours of the amplitude (cm s ) in the shelf edge region of the clockwise rotating component of current at (a) f, (b) M2, (c) M4 and (d) fM2 frequencies.

the clockwise component in the surface layer show regions of enhanced internal tide due to its propagation both onto the shelf and into the ocean. Due to self-interaction through the non-linear momentum terms involving horizontal currents and their horizontal derivatives, the M4 internal tide is generated from the M2 component (Fig. 14). Nonlinear interaction between inertial oscillations and the M2 internal tide produce energy at the fM2 frequency. However the spatial distribution of this is signiﬁcantly different from the M4 tide (see Fig. 14) with the fM2 frequency showing a maximum at

the region of the thermocline. A detailed analysis of the various contributions to the fM2 frequency showed (Xing and Davies, 2002; Davies and Xing, 2003b) that it was generated primarily from vertical shear in the inertial oscillations which was a maximum at the level of the thermocline, and vertical velocity due to the internal tide. These calculations show that the momentum advection terms have a signiﬁcant inﬂuence both upon inertial oscillations and near-inertial internal waves through both the background vorticity associated with a sheared steady ﬂow and through non-linear interaction due to time dependent processes.

Fig. 13. (a) Contours of the variance V at the surface computed with the non-linear prognostic model and an along shelf current to the north at days 4 and 10. (b) Contours of hðu huiÞ2 i along cross-section C3, for days 4 and 10. (c) Temperature contours at cross-section C3 for day 10.

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9. Concluding discussion The initial part of the paper presented a brief overview of the research which has been performed previously to examine internal wave generation and propagation. In the deep ocean, far away from coastal inﬂuences, divergences or convergences in the wind ﬁeld give rise to nearinertial internal wave production. In the case of a uniform wind and a linear model, inertial oscillations are uniform in the horizontal, with nearinertial energy conﬁned to a surface layer. Inclusion of a coastal boundary or shear in the background current in combination with nonlinear effects gives rise to spatial variability with resulting Ekman pumping producing internal waves at depth. In the coastal ocean, the presence of a coastline, gives rise to upwelling and downwelling associated with off-shore/on-shore wind induced ﬂows. At the coast these produce gradients of sea surface elevation, and an associated barotropic ﬂow (a gradient current) phase shifted by 1801 from the direct surface wind driven ﬂow (wind drift current). Calculations using the cross-sectional and single point models with sinusoidal wind forcing conﬁrm that away from the region of coastal inﬂuence ‘‘gradient’’ and ‘‘drift’’ components of the barotropic current explain the observed 1801 phase shift in current. The distribution of current magnitude between the surface layer and lower layer can also be explained by such a model. However, the single point model cannot explain the nearshore decay of surface nearinertial current amplitude, or the inﬂuence of internal pressure gradients in the near coastal region due to propagating internal waves. Calculations with the cross-sectional model show that at sub-inertial frequencies internal waves are trapped in the coastal region, while for super-inertial frequencies their offshore extent increases with increasing frequency. In the case of deepening of the thermocline in response to a wind pulse, the inability of the single point model to radiate energy through internal wave propagation leads to a different thermocline depth and intensity to that found in the cross-section model. Also single point models cannot take account of advection into a

region. This suggests that under stratiﬁed conditions the use of single point models is extremely limited. The need to adjust the minimum background turbulence energy within the thermocline in a single point model to take account of internal waves and other effects in order to determine the correct level of mixing (Burchard et al., 2002; Bolding et al., 2002) highlights the importance of the range of physical processes that cannot be included in these models. Indeed, Luyten et al. (2002) concluded that for inertial oscillations in the thermocline, due to episodic wind events, and the associated mixing, three-dimensional models are required. These models should have a sufﬁciently high horizontal resolution (order 1 to 2 km) to resolve the baroclinic Rossby radius. Also high temporal and spatial wind data needs to be available. In addition, Bolding et al. (2002) point to the inability of single point models to take account of advection effects, and in particular the short period vertical movements of the thermocline due to inﬂow/outﬂow into a region. The lack of these vertical movements signiﬁcantly inﬂuences the ability of the model to compute an appropriate mixed layer thickness. The importance of nearshore topography in determining the offshore propagation of internal waves was also demonstrated with the crosssectional model. Calculations with the crosssectional model also showed (Davies and Xing, 2002) that near coastal fronts non-linear effects were responsible for the offshore propagation of internal waves. The presence of the coastline produces a current at depth with an associated 1801 phase shift from the surface. Trapping of vertically propagating internal waves in frontal regions was responsible for additional mixing at depth. This suggests that in frontal regions it is not only additional turbulence due to shear associated with propagating internal waves, but turbulence due to wave trapping that must be taken into account. Although the cross-sectional model gave valuable insight into the processes inﬂuencing internal wave propagation, calculations with the threedimensional model showed the severe limitations of cross-sectional models in regions of varying

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along-shore topography. In particular the importance of alongshore pressure gradients in these cases, upon internal wave propagation. The inﬂuence of an alongshelf ﬂow through the background vorticity term also had a signiﬁcant effect upon the distribution of near-inertial energy in the shelf edge region. In addition a detailed comparison of three-dimensional, two-dimensional (crosssectional) (2D) and one-dimensional (a single point in the vertical) (1D) models of wind forced currents and internal waves in the Balearic Sea (Xing et al. 2004) has recently shown the severe limitations of the 1D and 2D models compared to a full three-dimensional model. This detailed comparison conﬁrms the conclusion of Luyten et al., (2002) that three-dimensional models are required in studies of currents and turbulence in stratiﬁed regions. In a ﬁnal series of calculations the effects of wave–wave interaction in the shelf edge region was illustrated by the cascade of energy from the internal M2 tide, and wind forced inertial oscillations into the fM2 frequency. This calculation together with the other calculations, clearly shows that lateral boundary layer mixing of tidal and wind-induced origin, together with its diffusion into the ocean will have a signiﬁcant inﬂuence on ocean mixing and circulation (Samelson, 1998; Spall, 2001). Certainly the importance of the nearinertial internal wave band (Garrett, 2001) is evident from the fact that it contains 50% of the energy of the internal wave ﬁeld. In addition, the major source of mixing and turbulence in the deep ocean is from wind forced motion. How this interacts with its environment determines if deep ocean or boundary layer mixing (Inall et al., 2000) is the main source of oceanic mixing, with results presented here emphasizing the importance of boundary layer mixing. A general conclusion from the literature and the calculations presented here concerning the role of models in stratiﬁed regions is that although single point models are useful, they are clearly limited, with cross-sectional models having an important role (Davies and Xing, 2002). However, for realistic simulations and comparison with observations three-dimensional models are required. In any three-dimensional simulation that can take

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account of fronts and internal waves, it is essential to ensure that a sufﬁciently ﬁne horizontal and vertical grid together with an accurate density advection scheme is used, to ensure that artiﬁcial diffusion is removed. Future work is likely to involve such models, in which the effects of internal wave mixing can be properly accounted for. Accurate and meaningful comparisons can then be made with ﬁeld data. The need to acquire comprehensive and accurate ﬁeld data to compare with these models will be a major task. In particular detailed measurements of temperature, salinity and currents together with associated turbulence proﬁles in regions of strong mixing will be required. The shelf edge region and areas of rough topography will be key locations for such measurements. Associated detailed measurements of internal wave spectra (e.g. Gemmrich and van Haren, 2002) will be particularly important. Spectral measurements in the near-inertial band, will be vital in determining if three dimensional models can correctly reproduce the internal waves and hence mixing in this energetic part of the spectrum. As shown by Samelson (1988) and Spall (2001) the correct inclusion of mixing in ocean circulation models has a major effect upon the computed circulation determined by these models, with associated implications for climate research.

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Modelling processes influencing wind-induced internal wave generation and propagation

Modelling processes influencing wind-induced internal wave generation and propagation

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