Baroclinic and topographic influences on the transport in western boundary currents

Abstract

One of the central unsolved theoretical problems of the large scale ocean circulation is concerned with explaining the very large transports measured in western boundary currents such as the Gulf Stream and the Kuroshio. The only theory up to now that can explain the size of these transports is that of non-linear recirculation in which the advective terms in the momentum equations became important near the western boundary. In this paper an alternative explanation is suggested. When bottom topography and baroclinic effects are included in a wind-driven ocean model it is shown that the western boundary current can have a transport larger than that predicted from the wind stress distribution even when the nonlinear advective terms are ignored. The explanation lies in the presence of pressure torques associated with bottom topography which can contribute to the vorticity balance in the same sense as the wind stress curl.

Three numerical experiments have been carried out to explore the nature of this process using a three dimensional numerical model. The first calculation is done for a baroclinic ocean of constant depth, the second for a homogeneous ocean with an idealized continental slope topography, and the third for a baroclinic ocean with the same continental slope topography. The nature of the vorticity balance and of the circulation around closed paths is examined in each case, and it is shown that bottom pressure torques lead to enhanced transport in the western boundary current only for the baroclinic case with variable depth.     

Observations of semidiurnal internal tidal currents off central Chile (36.6°S)

Observations of semidiurnal internal tidal currents from three moorings deployed on the continental shelf off central Chile during summer and winter of 2005 are reported. The spectra of the baroclinic currents showed large peaks at the semidiurnal band with a dominant counterclockwise rotation, which was consistent with internal wave activity. The amplitude of the barotropic tidal currents varied according to the spring–neap cycle following the sea level fluctuations. In contrast, the amplitudes of the internal tide showed high spatial-temporal variability not directly related to the spring–neap modulation. Near the middle of the continental shelf and near the coast (San Vicente Bay) the variance of the semidiurnal baroclinic current is larger than the variance of its barotropic counterpart. The vertical structure of the baroclinic tidal current fluctuations was similar to the structure of the first baroclinic internal wave mode. In general, in the three study sites the variance of the baroclinic current was larger near the surface and bottom and tended to show a minimum value at mid depths. Kinetic energy related to semidiurnal internal waves was larger in winter when stratification of the water column was stronger. During summer, upwelling and the decrease of freshwater input from nearby rivers reduced the vertical density stratification. The amplitude of the semidiurnal internal tide showed a tendency to be enhanced with increasing stratification as observed in other upwelling areas. The continental shelf break and submarine canyons, which limit the continental shelf in the alongshore direction, represent near-critical slopes for the semidiurnal period and are suggested to be the main internal tide generation sites in the study region.     

High-frequency vertical current observations in stratified seas

Although large-scale tidal and inertial motions dominate the kinetic energy and vertical current shear in shelf seas and ocean, short-scale internal waves at higher frequencies close to the local buoyancy frequency are of some interest for studying internal wave breaking and associated diapycnal mixing. Such waves near the upper limit of the inertio-gravity wave band are thought to have relatively short O (10²–10³ m) horizontal scales and to show mainly up- and downward motions, which contrasts with generally low aspect ratio large-scale ocean currents. Here, short-term vertical current (w) observations using moored acoustic Doppler current profiler (ADCP) are presented from a shelf sea, above a continental slope and from the open ocean. The observed w, with amplitudes between 0.015 and 0.05 m s⁻¹, all span a considerable part of the water column, which is not a small vertical scale O(water depth) or O (100–500 m, the maximum range of observations), with either 0 or π phase change. This implies that they actually represent internal waves of low vertical modes 1 or 2. Maximum amplitudes are found in layers of largest stratification, some in the main pycnocline bordering the frictional bottom boundary layer, suggesting a tidal source. These ‘pycnocline-w’ compose a regular train of (solitary) internal waves and linearly decrease to small values near surface and bottom.     

Some observations of internal wave current fluctuations at the shelf-edge and their implications for sediment transport

Observations of internal wave current fluctuations at a site on the European continental shelf are described. These have revealed current ‘pulses’ of regular tidal (M₂) phase which may be associated with internal tides generated at the shelf-edge. Current ‘pulses’ have been observed with amplitudes of 30 to 40 cm s^?1 superimposed on peak spring tidal currents of the order 60 to 70 cm s^?1. The measurements have shown that these fluctuations extended throughout the bottom mixed layer to within at least 2 m of the sea bed where they may play an important role in modifying sediment transport rates.     

Tidally generated internal waves in partially mixed estuaries

The generation of internal waves in the partially mixed estuaries is examined. The numerical experiments consider the barotropic tidal currents interacting with isolated obstacles in an open channel. The bottom boundary layer and longitudinal salinity gradient are included. Internal lee (arrested) waves are excited when the accelerating barotropic tidal current approaches the first-mode internal wave speed. The arrested waves are amplified, and are subsequently released when the decelerating tidal current falls below the first-mode internal wave speed. The power input from the barotropic tidal energy into internal wave energy is calculated. It is on the order of 10⁻² W/m², and is comparable to the estimated interior dissipation rate. This suggests that the tidally generated internal waves could be a significant energy source for mixing in the halocline.     

Tides off‐shore: Transition from California coastal to deep‐sea waters‡

Abstract

Tidal pressures and currents were measured with self‐contained capsules dropped to the sea floor for one month at distances of 175, 190, and 500 nautical miles from San Diego. These observations, together with a one‐week bottom pressure record by Filloux at 750 n miles, and three half‐week bottom current records by Isaacs et al, at intermediary distances, were analyzed for tidal components by cross‐correlation with a noise‐free reference time series. (For short records this method has some merit over classical tide analysis.) It was found that the tide decays seaward to e^‐1 times the coastal amplitude over a distance of order 1000 km for the semidiurnal species, slower for the diurnal species. Tidal currents turn counterclockwise, and are polarized with maximum flow parrallel to shore in the direction of tidal propagation (320°T) at local high tide. The current amplitude is roughly 2 cm/sec for the semidiurnal component, 1 cm/sec for the diurnal component. Superimposed baroclinic tidal currents lead to poor signal: noise ratios (between 1:1 and 10:1) for the barotropic currents. In contrast, the ratio is typically 1000:1 for the bottom pressures and generally exceeds that for coastal tide stations of comparable duration. Published I.H.B. tidal constants for exposed California coastal stations indicate “upshore” (towards 320°T) propagation at 140 m/sec for semidiurnal tides. 214 m/sec for diurnal tides.

To interpret these diverse observations, we have computed the dispersion laws for all possible rotationally‐gravitationally trapped waves against a straight coast with shelf. Trapped solutions are conveniently portrayed in terms of a parameter μ such that ? = sin μ = iu/v and f = ‐ cos μ = η/v define the ellipticity and impedance of the wave motion, η, u and v being off‐shelf dimensionless elevation, normal‐to‐shore and longshore components of velocity, respectively. We then attempt to fit the observations by a superposition of the possible wave classes, all of the same tidal frequency: (a) a free Kelvin‐like edge wave with small μ (mostly trapped by rotation, but somewhat slowed by the shelf); (6) a free Poincare‐like leaky wave; and (c) a forced wave (the distortion of the sea bottom by the tide producing forces plays a significant role). The mod el can account for the main features of the observed tidal heights, and gives relative amplitudes at the coast of 54:16:4 cm for components a:b:c in the case of the semidiurnal tides, 21:24:9 cm for the diurnal tides. The results place a semidiurnal amphidrome about midway between San Diego and Hawaii. Tidal currents are not well fitted by the model, and there are problems associated with the separation of barotropic and baroclinic modes, and with the benthic boundary layer. Coastal energy dissipation is small in the sea under investigation, but a “ capacitive “ phase delay appears to be associated with Northern California harbors and inland waters.     

Generation of baroclinic topographic waves by a tropical cyclone impacting a low-latitude continental shelf

Numerical model experiments have been performed to analyze the low-latitude baroclinic continental shelf response to a tropical cyclone. The theory of coastally trapped waves suggests that, provided appropriate slope, latitude, stratification and wind stress, bottom-intensified topographic Rossby waves can be generated by the storm. Based on a scale analysis, the Nicaragua Shelf is chosen to study propagating topographic waves excited by a storm, and a model domain is configured with simplified but similar geometry. The model is forced with wind stress representative of a hurricane translating slowly over the region at 6 km h⁻¹. Scale analysis leads to the assumption that baroclinic Kelvin wave modes have minimal effect on the low-frequency wave motions along the slope, and coastal-trapped waves are restricted to topographic Rossby waves. Analysis of the simulated motions suggests that the shallow part of the continental slope is under the influence of barotropic topographic wave motions and at the deeper part of the slope baroclinic topographic Rossby waves dominate the low-frequency motions. Numerical solutions are in a good agreement with theoretical scale analysis. Characteristics of the simulated baroclinic waves are calculated based on linear theory of bottom-intensified topographic Rossby waves. Simulated waves have periods ranging from 153 to 203 h. The length scale of the waves is from 59 to 87 km. Analysis of energy fluxes for a fixed volume on the slope reveals predominantly along-isobath energy propagation in the direction of the group velocity of a topographic Rossby wave. Another model experiment forced with a faster translating hurricane demonstrates that fast moving tropical cyclones do not excite energetic baroclinic topographic Rossby waves. Instead, robust inertial oscillations are identified over the slope.     

Bottom instrumented tripods: History,applications, and impacts

Instrumented bottom tripods have provided important data on sediment transport processes on continental shelves and in estuaries for four decades. Since the initial deployment in a tidal channel in Puget Sound, WA, in 1965 numerous tripods have been constructed to investigate bottom boundary layer and sediment dynamics worldwide. Tripod data have led to new understanding of near-bottom wave and current flows in the coastal ocean, and have been crucial to the development of shelf circulation and sediment transport models. Calculations of bottom stress, bottom roughness, and sediment flux that resulted directly from tripod data have been compared to bottom boundary layer model results. Where these have differed, new or revised model components have been developed to improve the skill of the models. The many discoveries that have been made from tripod experiments include dense, near-bottom fluid mud layers that transport large quantities of suspended sediment offshore into deeper regions of the continental shelf. This process has been linked to the seaward progradation of subaqueous deltas and to the boundaries of mid-shelf mud deposits off rivers with high fine-sediment discharge.     

“The Ekman Drain”: a conduit to the deep ocean for shelf material

A long (167 days) acoustic Doppler current profiler time series from the European continental slope west of Scotland has been analysed to investigate the influence of bathymetric steering on the slope current and the extent of down-slope transport in the bottom boundary layer. Within an interior region between the surface and bottom boundary layers, the direction of the flow is found to be remarkably consistent as required by the Taylor-Proudman theorem for geostrophic flow. The mean value of this interior flow direction is taken to be the effective direction of the bathymetry in controlling the geostrophic flow and so defines the rotation of coordinates required to determine along and cross-flow transports. Within a bottom boundary layer (BBL) of thickness ~100 m, the direction of the flow was deflected increasingly to the left with the mean veering angle ~12.5° at 12 mab and a down-slope speed of 2.6 cm s^?1. The corresponding integrated transport (the “Ekman drain”) had an average value of ~1.6 m² s^?1 over the full observation period. This down-slope flow was significantly correlated (at 0.1 % level), with the stress applied by the along-slope flow although with considerable scatter (r.m.s. ~1 m² s^?1) which suggests the influence of other forcing mechanisms. Combining the BBL volume transport with an estimate of the mean concentration of suspended particulate material indicates an annual down-slope flux of 3.0?±?0.6 tonnes m^?1 year^?1, of which ~0.36?±?0.1 tonnes m^?1 year^?1 is carbon. Biogeochemical measurements indicate that the carbon flux in the Ekman drain predominates over settlement of organic material through the water column over the slope and provides for relatively rapid delivery of material to deep water.     

Tidal-induced buoyancy flux and mean transverse circulation

A weakly nonlinear model is used to examine the mean transverse circulation (cross-isobath) driven by tidal-induced buoyancy flux. The mean Eulerian flows driven by both the barotropic and baroclinic tide are presented for a semi-infinite wedge. The mean flow driven by the barotropic tide is significant only near the apex where the thickness of the frictional boundary layer is comparable to the water depth. The mean flow there is characterized by a single-cell circulation with offshore flow near the bottom, and its magnitude can reach a few percentage or a significant fraction of the tidal velocity in oceanic applications. The mean flow driven by the baroclinic tide, on the other hand, is characterized by pairs of half-open (on the seaward side) counter-rotating cells, the number of which equals the vertical mode number. For a baroclinic tide propagating onshore, the mean flow near the top and bottom surfaces is always directed offshore and its magnitude can reach a large fraction of the tidal velocity. Taken together, the model thus predicts a mean offshore flow near the bottom while higher up in the water column the mean flow direction is less definite due to the contribution from different tidal components. The model results are consistent with some current measurements over the Georges Bank.     

Sill dynamics and energy transformation in a jet fjord

A detailed set of observations are presented of the tidal forcing and basin response of Loch Etive, a jet-type fjordic system on the west coast of Scotland. The characteristics of the tidal jet observed during a spring tide are discussed in detail, and with reference to laboratory studies of Baines and Hoinka (1985). Although the system is categorized as a jet basin during spring tides (when the mode-1 densimetric Froude number exceeds 1) and a wave basin during neap tides (when the Froude number remains below 1), a mode-1 baroclinic wave response is observed throughout the spring/neap cycle. Of the total incident tidal energy, 16% is lost from the barotropic tide. The ratio between loss to bottom friction, barotropic form drag and baroclinic wave drag is estimated to be 1:4:1 (1:4:3.3) at springs (neaps). Despite this, during a spring tide, a 20-m amplitude baroclinic mode-1 wave is observed to propagate along the full length of the basin at a speed of 0.2 m s^–1, somewhat slower than the predicted linear mode-1 phase speed. A hydrographic section supports the implication of the dissipation of the baroclinic wave towards the loch head. The stratification of the upper layers is observed to decrease rapidly landward of the 40-m isobath, a possible signature of enhanced diapycnal mixing in the shallower reaches towards the loch head.Responsible Editor: Jens Kappenberg     

A test of the influence of tidal stream polarity on the structure of turbulent dissipation

Vertical mixing by the tides plays a key role in controlling water column structure over the seasonal cycle in shelf seas. The influence of tidal stirring is generally well represented as a competition between surface buoyancy input and the production of turbulent kinetic energy (TKE) by frictional stresses, a competition which is encapsulated in the Qh/u³ criterion. An alternative control mechanism arises from the limitation of the thickness of the bottom boundary layer due to the effects of rotation and the oscillation of the flow. Model studies indicate that, for conditions typical of the European shelf seas, the energy constraint exerts the dominant control but that for tidal streams with large positive polarisation (i.e. anti-clockwise rotation of velocity vector), some influence of rotation in limiting mixing should be detectable. We report here measurements of flow structure (with ADCPs) and turbulent dissipation (FLY Profiler) made at two similar locations in the Celtic Sea which differ principally in that the tidal currents rotate in opposite senses with approximately equal magnitude (polarity P=±0.6). A clear contrast was observed between the two sites in the vertical structure of the currents, the density profile and the rate of dissipation of TKE. At the positive polarity (PP) site (P≈+0.6), the bottom boundary layer in the tidal flow was limited to ∼20 mab (metre above the bed) and significant dissipation from bottom boundary friction was constrained within this layer. At the negative polarity (NP) site (P≈−0.6), the dominant clockwise rotary current component exhibited a velocity defect (i.e. reduction relative to the free stream) extending into the upper half of the water column while significant dissipation was observed to penetrate much further up the water column with dissipation levels ∼10^−4.5 W m⁻³ reaching to the base of the pycnocline at 70–80 mab. These contrasting features of the vertical distribution of dissipation are well reproduced by a 1-D model when run with windstress and tidal forcing and using the observed density profile. Model runs with reversed polarity at the two sites, support the conclusion that the observed contrast in the structure of tidal velocity, dissipation and stratification is due to the influence of tidal stream polarity. Increased positive polarity reduces the upward penetration of mixing which allows the development of stronger seasonal stratification, which, in turn, further inhibits vertical mixing.     

Observations of the spring–neap modulation of the gravitational circulation in a partially mixed estuary

The long-term variability of the non-tidal circulation in Southampton Water, a partially mixed estuary, was investigated using 71-day acoustic Doppler current profiler (ADCP) time series. The data show evidence that the spring–neap tidal variability of the turbulent mixing modulates the strength of the non-tidal residual circulation, with subtidal neap tide surface flows reaching 0.12 m s^–1 compared to <0.05 m s^–1 at spring tides. The amplitude of the neap-tide events in this non-tidal circulation is shown to be related to a critical value of the tidal currents, illustrating the strong dependence on tidal mixing. The results suggest that the dominant mechanism for generating these neap-tide circulation events is the baroclinic forcing of the horizontal density gradient, rather than barotropic forcing associated with ebb-induced periodic stratification. While tidal turbulence is thought to be the dominant control on this gravitational circulation, there is evidence of the additional effect of wind-driven mixing, including the effects of wind fetch and possibly wave development with along-estuary winds being more efficient at mixing the estuary than across-estuary winds. Rapid changes in atmospheric pressure also coincided with fluctuations in the gravitational circulation. The observed subtidal flows are shown to be capable of rapidly flushing buoyant material out of the estuary and into the coastal sea at neap tides.Responsible Editor: Iris Grabemann     

A numerical study of island wake generated by an elliptical tidal flow

An idealized numerical study of the influence of a tidal flow around an island has been undertaken with ROMS. The study focusses on coastal island wakes which are mainly controlled by elliptical tidal current flows on shallow shelves. This model is typical of some isolated continental shelf islands. The model is forced by a semi-diurnal barotropic inertia gravity wave imposed on the four open boundaries of a rectangular domain and its propagation results in an elliptical tidal flow within the domain in which the circular island lies. The influence of the surrounding island bathymetry and of the ellipse shape has been studied both in two and three dimensions. In the island vicinity, the residual circulation patterns over a tidal period show alongshore flow divergence along the major axis and convergence along the minor axis. A thin tidal ellipse (i.e. with a large ratio between major and minor axes) leads to strong eddy activity periods in the lee of the island during the flood and ebb phases, with eddy dissipation phases in between. By contrast, an almost round ellipse (axis ratio nearly 1) leads to vorticity filaments which continuously progress around the island without eddy shedding. The presence of a topographic slope in the vicinity of the island strengthens the eddy activity. This study suggests that the tidal current rotation favors the development of the eddy rotating in the same direction and weakens the development of the second eddy. In three dimensions with a surrounding bathymetry, an intense upwelling occurs in a large area in the lee of the island and the vertical velocities are stronger with thinner ellipses. With a flat bottom the vertical motions are almost fully generated by convergence and divergence of the secondary flow. With a varying bottom topography, the vertical motions come from a combination of this mechanism with convergence and divergence of the depth averaged flow.     

The impact of the spatial variability in bottom roughness on tidal dynamics and energetics, a case study: the M 2 surface tide in the North European Basin

A modified version of the 3D finite-element hydrostatic model QUODDY-4 is used to quantify the changes in the dynamics and energetics of the M ₂ surface tide in the North European Basin, induced by the spatial variability in bottom roughness. This version differs from the original one, as it introduces a module providing evaluation of the drag coefficient in the bottom boundary layer (BBL) and by accounting for the equilibrium tide. The drag coefficient is found from the resistance laws for an oscillatory rotating turbulent BBL over hydrodynamically rough and incompletely rough underlying surfaces, describing how the wave friction factor as well as other resistance characteristics depend on the dimensionless similarity parameters for the BBL. It is shown that the influence of the spatial variability in bottom roughness is responsible for some specific changes in the tidal amplitudes, phases, and the maximum tidal velocities. These changes are within the model noise, while the changes in the averaged (over a tidal cycle) horizontal wave transport and the averaged dissipation of barotropic tidal energy may be of the same orders of magnitude as are the above energetic characteristics as such. Thus, contrary to present views, ignoring the spatial variability in bottom roughness at least in the North European Basin is only partially correct: it is valid for the tidal dynamics, but is liable to break down for the tidal energetics.     

Some hydrodynamic problems for a nepheloid zone

Summary A layer of a few hundred meters thickness with suspended matter (a nepheloid zone) was discovered byEwing andThorndike [4]³⁾ near the bottom on the continental slope of the North Atlantic. A downward pressure gradient is produced in this layer due to increment of water density with suspensoid. When only the Coriolis force balances with this pressure gradient, a bottom nepheloid current flows southwestward parallel to the depth contours with a velocity of about 10 (cm/sec) for a slope of one degree. The pressure gradient for fluid with locally variable density above a sloping bottom is treated and an extra term due to density gradient along the slope is derived. The vertical profiles of the nepheloid current with an effect on the vertical eddy viscosity are computed. Two kinds of vertical distributions of eddy viscosity are determined from the observed nepheloid distributions and used in the calculations: constant but different values at two layers and those increasing with height. The effect of the change of density along the bottom is treated by introducing dimensionless variables. Rossby number of the nepheloid current becomes about 10^–2 indicating inertia terms to be negligible. Rossby number of turbidity currents ranges from 2 (in a decaying area) to 5 (developing area), suggesting that inertia terms are more important than Coriolis terms. The trajectories of turbidity currents are computed from motion of a mass of mud under the Coriolis force and friction, and the results are applied to those inferred byHand andEmery [6] in the San Diego Through off California.LGO Contribution Number 925.     

Successive formation of planetary lenses in an intermediate layer

Abstract

We study the formation of lenses of the ocean's intermediate water using a 2.5-layerβ-plane primitive equation model with localized injection of water mass. For the injecting rate of 1.0 Sv, we have observed that strong vortices are shed regularly. These vortices propagate westward much faster than the second baroclinic long Rossby wave. They are totally isolated from each other and show strong baroclinicity as well. Moreover, they remain stable over a sufficiently long period of time. Regular formation of such strong vortices in the intermediate layer has not been reported previously. The translation speed is explained using the Euler's momentum integral theorem for the nonlinear baroclinic vortex on the β-plane. We have demonstrated that coupling between the primary motion in the intermediate layer and the secondary motion in the upper layer with a meridional shift is crucial to the fast westward translation of the intense vortices. A simple dispersion formula relating the zonal translation speed with the vortex radius is also derived under the assumption of quasi-geostrophy. It has turned out that the analytical relation explains the numerical results surprisingly well despite the limitation of its derivation.     

Boundary layer dissipation and the nonlinear interaction of equatorial baroclinic and barotropic rossby waves

Two-layer equatorial primitive equations for the free troposphere in the presence of a thin atmospheric boundary layer and thermal dissipation are developed here. An asymptotic theory for the resonant nonlinear interaction of long equatorial baroclinic and barotropic Rossby waves is derived in the presence of such dissipation. In this model, a self-consistent asymptotic derivation establishes that boundary layer flows are generated by meridional pressure gradients in the lower troposphere and give rise to degenerate equatorial Ekman friction. That is to say, the asymptotic model has the property that the dissipation matrix has one eigenvalue which is nearly zero: therefore the dynamics rapidly dissipates flows with pressure at the base of the troposphere and creates barotropic/baroclinic spin up/spin down. The simplified asymptotic equations for the amplitudes of the dissipative equatorial barotropic and baroclinic waves are studied by linear theory and integrated numerically. The results indicate that although the dissipation slightly weakens the tropics to midlatitude connection, strong localized wave packets are nonetheless able to exchange energy between barotropic and baroclinic waves on intraseasonal timescales in the presence of baroclinic mean shear. Interesting dissipation balanced wave-mean flow states are discovered through numerical simulations. In general, the boundary layer dissipation is very efficient for flows in which the barotropic and baroclinic components are of the same sign at the base of the free troposphere whereas the boundary layer dissipation is less efficient for flows whose barotropic and baroclinic components are of opposite sign at the base of the free troposphere.     

The impact of the spatial variability in bottom roughness on tidal dynamics and energetics,a case study: the M 2 surface tide in the North European Basin

A modified version of the 3D finite-element hydrostatic model QUODDY-4 is used to quantify the changes in the dynamics and energetics of the M ₂ surface tide in the North European Basin, induced by the spatial variability in bottom roughness. This version differs from the original one, as it introduces a module providing evaluation of the drag coefficient in the bottom boundary layer (BBL) and by accounting for the equilibrium tide. The drag coefficient is found from the resistance laws for an oscillatory rotating turbulent BBL over hydrodynamically rough and incompletely rough underlying surfaces, describing how the wave friction factor as well as other resistance characteristics depend on the dimensionless similarity parameters for the BBL. It is shown that the influence of the spatial variability in bottom roughness is responsible for some specific changes in the tidal amplitudes, phases, and the maximum tidal velocities. These changes are within the model noise, while the changes in the averaged (over a tidal cycle) horizontal wave transport and the averaged dissipation of barotropic tidal energy may be of the same orders of magnitude as are the above energetic characteristics as such. Thus, contrary to present views, ignoring the spatial variability in bottom roughness at least in the North European Basin is only partially correct: it is valid for the tidal dynamics, but is liable to break down for the tidal energetics.