Nonlinear tides are generated by hydrodynamic distortions of tides. A tidal wave is said to be nonlinear when its shape deviates from a pure sinusoidal wave. In mathematical terms, the wave owes its nonlinearity due to the nonlinear advection and frictional terms in the governing equations. These become more important in shallow-water regions such as in estuaries. Nonlinear tides are studied in the fields of coastal morphodynamics, coastal engineering and physical oceanography. The nonlinearity of tides has important implications for the transport of sediment.
Framework edit
From a mathematical perspective, the nonlinearity of tides originates from the nonlinear terms present in the Navier-Stokes equations. In order to analyse tides, it is more practical to consider the depth-averaged shallow water equations:[1]
These equations follow from the assumptions that water is incompressible, that water does not cross the bottom or surface and that pressure variations above the surface are negligible. The latter allows the pressure gradient terms in the standard Navier-Stokes equations to be replaced by gradients in . Furthermore, the coriolis and molecular mixing terms are omitted in the equations above since they are relatively small at the temporal and spatial scale of tides in shallow waters.
For didactic purposes, the remainder of this article only considers a one-dimensional flow with a propagating tidal wave in the positive -direction.This implies that zero and is all quatities are homogeneous in the -direction. Therefore, all terms equal zero and the latter of the above equations is arbitrary.
Nonlinear contributions edit
In this one dimensional case, the nonlinear tides are induced by three nonlinear terms. That is, the divergence term , the advection term , and the frictional term . The latter is nonlinear in two ways. Firstly, because is (nearly) quadratic in . Secondly, because of in the denominator. The effect of the advection and divergence term, and the frictional term are analysed separately. Additionally, nonlinear effects of basin topography, such as intertidal area and flow curvature can induce specific kinds of nonlinearity. Furthermore, mean flow, e.g. by river discharge, may alter the effects of tidal deformation processes.
Harmonic analysis edit
A tidal wave can often be described as a sum of harmonic waves. The principal tide (1st harmonic) refers to the wave which is induced by a tidal force, for example the diurnal or semi-diurnal tide. The latter is often referred to as the tide and will be used throughout the remainder of this article as the principal tide. The higher harmonics in a tidal signal are generated by nonlinear effects. Thus, harmonic analysis is used as a tool to understand the effect the nonlinear deformation. One could say that the deformation dissipates energy from the principal tide to its higher harmonics. For the sake of consistency, higher harmonics having a frequency that is an even or odd multiple of the principle tide may be referred to as the even or odd higher harmonics respectively.
Divergence and advection edit
In order to understand the nonlinearity induced by the divergence term, one could consider the propagation speed of a shallow water wave.[2] Neglecting friction, the wave speed is given as:[3]
Comparing low water (LW) to high water (HW) levels ( ), the through (LW) of a shallow water wave travels slower than the crest (HW). As a result, the crest "catches up" with the trough and a tidal wave becomes asymmetric.[4]
In order to understand the nonlinearity induced by the advection term, one could consider the amplitude of the tidal current.[2] Neglecting friction, the tidal current amplitude is given as:
When the tidal range is not small compared to the water depth, i.e. is significant, the flow velocity is not negligible with respect to . Thus, wave propagation speed at the crest is while at the trough, the wave speed is . Similar to the deformation induced by the divergence term, this results in a crest "catching up" with the trough such that the tidal wave becomes asymmetric.
For both the nonlinear divergence and advection term, the deformation is asymmetric. This implies that even higher harmonics are generated, which are asymmetric around the node of the principal tide.
Mathematical analysis edit
The linearized shallow water equations are based on the assumption that the amplitude of the sea level variations are much smaller than the overall depth.[1] This assumption does not necessarily hold in shallow water regions. When neglecting the friction, the nonlinear one-dimensional shallow water equations read:
Here .
When inserting this linear series in the nondimensional governing equations, the zero-order terms are governed by:
Collecting the terms and dividing by yields:
This linear inhomogenous partial differential equation, obeys the following particulate solution:
Returning to the dimensional solution for the sea surface elevation:
This solution is valid for a first order perturbation. The nonlinear terms are responsible for creating a higher harmonic signal with double the frequency of the principal tide. Furthermore, the higher harmonic term scales with , and . Hence, the shape of the wave will deviate more and more from its original shape when propagating in the -direction, for a relatively large tidal range and for shorter wavelengths. When considering a common principal tide, the nonlinear terms in the equation lead to the generation of the harmonic. When considering higher-order terms, one would also find higher harmonics.
Friction edit
The frictional term in the shallow water equations, is nonlinear in both the velocity and water depth.
In order to understand the latter, one can infer from the term that the friction is strongest for lower water levels. Therefore, the crest "catches up" with the trough because it experiences less friction to slow it down. Similar to the nonlinearity induced by the divergence and advection term, this causes an asymmetrical tidal wave.
In order to understand the nonlinear effect of the velocity, one should consider that the bottom stress is often parametrized quadratically:
Twice per tidal cycle, at peak flood and peak ebb, reaches a maximum, . However, the sign of is opposite for these two moments. Causally, the flow is altered symmetrical around the wave node. This leads to the conclusion that this nonlinearity results in odd higher harmonics, which are symmetric around the node of the principal tide.
Mathematical analysis edit
Nonlinearity in velocity edit
The parametrization of contains the product of the velocity vector with its magnitude. At a fixed location, a principal tide is considered with a flow velocity:
Here, is the flow velocity amplitude and is the angular frequency. To investigate the effect of bottom friction on the velocity, the friction parameterization can be developed into a Fourier series:
This shows that can be described as a Fourier series containing only odd multiples of the principal tide with frequency . Hence, the frictional force causes an energy dissipation of the principal tide towards higher harmonics. In the two dimensional case, also even harmonics are possible.[5] The above equation for implies that the magnitude of the friction is proportional to the velocity amplitude . Meaning that stronger currents experience more friction and thus more tidal deformation. In shallow waters, higher currents are required to accommodate for sea surface elevation change, causing more energy dissipation to odd higher harmonics of the principal tide.
Nonlinearity in water depth edit
Although not very accurate, one can use a linear parameterization of the bottom stress:[6]
Here is a friction factor which represents the first Fourier component of the more exact quadratical parameterization. Neglecting the advectional term and using the linear parameterization in the frictional term, the nondimensional governing equations read:
In a similar manner, the equations can be determined:
Intertidal area edit
In a shallow estuary, nonlinear terms play an important role and might cause tidal asymmetry. This can intuitively be understood when considering that if the water depth is smaller, the friction slows down the tidal wave more. For an estuary with small intertidal area (case i), the average water depth generally increases during the rising tide. Therefore, the crest of the tidal wave experiences less friction to slow it down and it catches up with the trough. This causes tidal asymmetry with a relatively fast rising tide. For an estuary with much intertidal area (case ii), the water depth in the main channel also increases during the rising tide. However, because of the intertidal area, the width averaged water depth generally deceases. Therefore, the trough of the tidal wave experiences relatively little friction slowing it down and it catches up on the crest. This causes tidal asymmetry with a relatively slow rising tide. For a friction dominated estuary, the flood phase corresponds to the rising tide and the ebb phase corresponds to the falling tide. Therefore, case (i) and (ii) correspond to a flood and ebb dominated tide respectively.
In order to find a mathematical expression to find the type of asymmetry in an estuary, the wave speed should be considered. Following a non-linear perturbation analysis,[7] the time-dependent wave speed for a convergent estuary is given as:[8]
With the channel depth, the estuary width, and the right side just a decomposition of these quantities in their tidal averages (denote by the ) and their deviation from it. Using a first order Taylor expansion, this can be simplified to:
Here:
This parameter represents the tidal asymmetry. The discussed case (i), i.e. fast rising tide, corresponds to , while case (ii), i.e. slow rising tide, corresponds to . Nonlinear numerical simulations by Friedrichs and Aubrey[9] reproduce a similar relationship for .
Flow curvature edit
Consider a tidal flow induced by a tidal force in the x-direction such as in the figure. Far away from the coast, the flow will be in the x-direction only. Since at the coast the water cannot flow cross-shore, the streamlines are parallel to the coast. Therefore, the flow curves around the coast. The centripetal force to accommodate for this change in the momentum budget is the pressure gradient perpendicular to a streamline. This is induced by a gradient in the sea level height.[10] Analogues to the gravity force that keeps planets in their orbit, the gradient in sea level height for a streamline curvature with radius is given as:
For a convex coast, this corresponds to a decreasing water level height when approaching the coast. For a concave coast this is opposite, such that the sea level height increases when approaching the coast. This pattern is the same when the tide reverses the current. Therefore, one finds that the flow curvature lowers or raises the water level height twice per tidal cycle. Hence it adds a tidal constituent with a frequency twice that of the principal component. This higher harmonic is indicative of nonlinearity, but this is also observed by the quadratic term in the above expression.
Mean flow edit
A mean flow, e.g. a river flow, can alter the nonlinear effects. Considering a river inflow into an estuary, the river flow will cause a decrease of the flood flow velocities, while increasing the ebb flow velocities. Since the friction scales quadratically with the flow velocities, the increase in friction is larger for the ebb flow velocities than the decrease for the flood flow velocities. Hence, creating a higher harmonic with double the frequency of the principal tide. When the mean flow is larger than the amplitude of the tidal current, this would lead to no reversal of the flow direction. Thus, the generation of the odd higher harmonics by the nonlinearity in the friction would be reduced. Moreover, an increase in the mean flow discharge can cause an increase in the mean water depth and therefore reduce the relative importance of nonlinear deformation.[11]
Example: Severn Estuary edit
The Severn Estuary is relatively shallow and its tidal range is relatively large. Therefore, nonlinear tidal deformation is notable in this estuary. Using GESLA data [1] of the water level height at the measuring station near Avonmouth, the presence of nonlinear tides can be confirmed. Using a simple harmonic fitting algorithm with a moving time window of 25 hours, the water level amplitude of different tidal constituents can be found. For 2011, this has been done for the , and constituents. In the figure, the water level amplitude of the and harmonics, and respectively, are plotted against the water level amplitude of the principal tide, . It can be observed that higher harmonics, being generated by nonlinearity, are significant with respect to the principal tide.
The correlation between and looks somewhat quadratic. This quadratic dependence could be expected from the mathematical analysis in this article. Firstly, the analysis of divergence and advection results in an expression that, for a fixed , implies:
Secondly, the analysis of the nonlinearity of the friction in the water depth yields a second higher harmonic. For the mathematical analysis, a linear parameterization of the bottom stress was assumed. However, the bottom stress actually scales nearly quadratically with the flow velocity. This is reflected in the quadratic relation between and .
In the graph, for a small tidal range, the correlation between and is approximately directly proportional. This relation between the principal tide and its third harmonic follows from the nonlinearity of the friction in the velocity, which is reflected in the derived expression. For larger tidal ranges, start decreasing. This behaviour remains unresolved by the theory covered in this article.
Sediment transport edit
The deformation of tides can be of significant importance in sediment transport.[15] In order to analyse this, it is obvious to distinguish between the dynamics of suspended sediment and bed load sediment. Suspended sediment transport (in one dimension) can in general be quantified as:[16]
Here is the depth integrated sediment flux, is the sediment concentration, is the horizontal diffusivity coefficient and is the reference height above the surface . The bed load transport can be estimated by the following heuristic definition:
The zonal flow velocity can be represented as a truncated Fourier series. When considering a tidal flow composed of only and constituents, the current at a specific location is given as:
Here is the fall velocity, is the vertical diffusivity coefficient and is an erosion coefficient. Advection is neglected in this model. Considering the definition of and , an expression for the tidally averaged bed and suspended load transport can be obtained:
Velocity asymmetry edit
The velocity asymmetry mechanism relies on a difference in maximum flow velocity between peak ebb and flood. The quantification of this mechanism is encapsulated by the term. The implications of this term are summarized in the table below:
0 | 0 | |
Hence, the velocity asymmetry mechanism causes a net ebb directed transport if the absolute value of the relative phase difference , while it causes a net flood directed transport if . In the latter case, peak flood flows will be larger than peak ebb flows. Hence, the sediment will be transported over a larger distance in the flood direction, making and . The opposite applies for .
Duration asymmetry edit
The duration asymmetry mechanism can also cause a tidally averaged suspended load transport. This mechanism only allows for a tidally averaged suspended sediment flux. The quantification of this mechanism is encapsulated by the term, which is absent in the equation. The implications of this term are summarized in the table below:
0 | |
When , the time from peak flood to peak ebb is longer than the time from peak ebb to peak flood. This makes that more sediment can settle during the period from peak flood to peak ebb, hence less sediment will be suspended at peak ebb and there will be a net transport in the flood direction. A similar, but opposite explanation holds for . Bed load transport is not affected by this mechanism because the mechanism requires a settling lag of the particles, i.e. the particles must take time to settle and the concentration adapts gradually to the flow velocities.
See also edit
References edit
- ^ a b Cushman-Roisin, Benoit; Beckers, Jean-Marie (2011). Introduction to geophysical fluid dynamics: physical and numerical aspects (2nd ed.). Waltham, MA: Academic Press. ISBN 978-0-12-088759-0. OCLC 760173075.
- ^ a b B., Parker, Bruce (1991). Tidal hydrodynamics. Wiley. ISBN 0-471-51498-5. OCLC 231330044.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - ^ Pond, Stephen (1991). Introductory dynamical oceanography. George L. Pickard (Second ed.). Oxford. ISBN 978-0-08-057054-9. OCLC 886407149.
{{cite book}}
: CS1 maint: location missing publisher (link) - ^ Dronkers, J. (1986-08-01). "Tidal asymmetry and estuarine morphology". Netherlands Journal of Sea Research. 20 (2): 117–131. Bibcode:1986NJSR...20..117D. doi:10.1016/0077-7579(86)90036-0. ISSN 0077-7579.
- ^ Pugh, David; Woodworth, Philip (2014). Sea-Level Science: Understanding Tides, Surges, Tsunamis and Mean Sea-Level Changes. Cambridge: Cambridge University Press. doi:10.1017/cbo9781139235778. ISBN 978-1-107-02819-7.
- ^ Friedrichs, Carl T. (2010), Valle-Levinson, Arnoldo (ed.), "Barotropic tides in channelized estuaries", Contemporary Issues in Estuarine Physics, Cambridge: Cambridge University Press, pp. 27–61, doi:10.1017/cbo9780511676567.004, ISBN 978-0-511-67656-7, retrieved 2022-03-20
- ^ Friedrichs, Carl T.; Madsen, Ole S. (1992). "Nonlinear diffusion of the tidal signal in frictionally dominated embayments". Journal of Geophysical Research. 97 (C4): 5637. Bibcode:1992JGR....97.5637F. doi:10.1029/92jc00354. ISSN 0148-0227.
- ^ Contemporary issues in estuarine physics. A. Valle-Levinson. Cambridge: Cambridge University Press. 2010. ISBN 978-0-511-67776-2. OCLC 648754476.
{{cite book}}
: CS1 maint: others (link) - ^ Friedrichs, Carl T.; Aubrey, David G. (November 1988). "Non-linear tidal distortion in shallow well-mixed estuaries: a synthesis". Estuarine, Coastal and Shelf Science. 27 (5): 521–545. Bibcode:1988ECSS...27..521F. doi:10.1016/0272-7714(88)90082-0. ISSN 0272-7714. S2CID 51119057.
- ^ Pugh, David (2014). Sea-level science : understanding tides, surges, tsunamis and mean sea-level changes. Philip Woodworth. Cambridge. pp. 135–136. ISBN 978-1-107-02819-7. OCLC 868079159.
{{cite book}}
: CS1 maint: location missing publisher (link) - ^ Tidal hydrodynamics. Bruce B. Parker. New York: J. Wiley. 1991. ISBN 0-471-51498-5. OCLC 23766414.
{{cite book}}
: CS1 maint: others (link) - ^ Haigh, Ivan; Marcos, Marta; Talke, Stefan; Woodworth, Philip; Hunter, John; Haugh, Ben; Arns, Arne; Bradshaw, Elizabeth; Thompson, Phil (2021-11-05). "GESLA Version 3: A major update to the global higher-frequency sea-level dataset". Eartharxiv Preprint. Bibcode:2021EaArX...X5MP65H. doi:10.31223/x5mp65. hdl:10261/353363. S2CID 243811785.
- ^ Woodworth, Philip L; Hunter, John R; Marcos Moreno, Marta; Caldwell, Patrick C; Menendez, Melisa; Haigh, Ivan David (2016), GESLA (Global Extreme Sea Level Analysis) high frequency sea level dataset - Version 2., British Oceanographic Data Centre, Natural Environment Research Council, doi:10.5285/3b602f74-8374-1e90-e053-6c86abc08d39, retrieved 2022-03-21
- ^ Caldwell, P. C.; Merrifield, M. A.; Thompson, P. R. (2001), Sea level measured by tide gauges from global oceans as part of the Joint Archive for Sea Level (JASL) since 1846, NOAA National Centers for Environmental Information, doi:10.7289/v5v40s7w, retrieved 2022-03-21
- ^ Dalrymple, Robert W.; Choi, Kyungsik (1978), "Sediment transport by tides", Sedimentology, Berlin, Heidelberg: Springer, pp. 993–998, doi:10.1007/3-540-31079-7_181, ISBN 978-3-540-31079-2, retrieved 2022-03-17
- ^ de Swart, H.e.; Zimmerman, J.t.f. (2009-01-01). "Morphodynamics of Tidal Inlet Systems". Annual Review of Fluid Mechanics. 41 (1): 203–229. Bibcode:2009AnRFM..41..203D. doi:10.1146/annurev.fluid.010908.165159. ISSN 0066-4189.
- ^ Groen, P. (1967-12-01). "On the residual transport of suspended matter by an alternating tidal current". Netherlands Journal of Sea Research. 3 (4): 564–574. Bibcode:1967NJSR....3..564G. doi:10.1016/0077-7579(67)90004-X. ISSN 0077-7579.