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Wave ripple formation
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{{Review
 
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|name=Job Dronkers|AuthorID=120|
 
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==Introduction==
 
 
 
[[Image: WaveRippleFormationFig0.jpg|thumb|350px|right|Figure 1. Ripples observed at Sea Rim State Park, along the coast of east Texas close to the border with Louisiana (courtesy by Zoltan Sylvester).]]
 
 
 
When a sandy seabed is subject to wave action and the wave orbital motion is strong enough to move sand grains, ripples often appear. The ripples induced by wave action are called “wave ripples”; their characteristics being different from those of the ripples generated by steady flows.
 
The most striking difference between wave ripple fields and current ripple fields is the regularity of the former. Indeed, regular long-crested wave ripple fields are often observed on tidal beaches from which the sea has withdrawn at low water (see figure 1).
 
 
 
The characteristics of wave ripples are described in the accompanying article [[Wave ripples]], which also explains their crucial importance for sand transport in the coastal zone (see also the articles [[Sand transport]] and [[Sediment transport formulas for the coastal environment]]).
 
 
 
The formation of wave ripple fields, characterized by well-defined wavelengths, is initiated by the interaction of seabed disturbances with the wave orbital motion. This article deals with the underlying processes giving rise to different wave ripple patterns.
 
 
 
 
 
 
 
 
 
==The mechanism of ripple formation==
 
 
 
Wave ripples form because the interaction of the oscillatory flow, induced by surface waves close to the bottom, with a bottom undulation of small amplitude generates steady streamings which consist of recirculating cells (see the flow visualizations of Kaneko and Honji<ref> Kaneko A. and Honji H. (1979.) Double structures of steady streaming in the oscillatory viscous flow over a wavy wall. J. Fluid Mech. 93, 727-736.</ref>). Indeed the presence of the bottom waviness induces periodic spatial variations of the streamwise oscillatory velocity component. Hence the nonlinear self-interaction of the velocity field, that is due to the convective acceleration, generates time-independent terms into momentum equation which can be balanced only by the presence of a steady velocity component.
 
 
 
[[Image: WaveRippleFormationFig1.jpg|thumb|400px|right|Figure 2. Steady streaming generated by an oscillatory flow over a wavy wall for <math>R_\delta=0.1</math>, <math>2 \pi \delta/\lambda=0.15</math> being the dimensionless wavenumber of the bottom waviness (adapted from Blondeaux<ref name=B90></ref>). In the figure <math>x</math> and <math>y</math> are dimensioness spatial coordinates scaled with <math>\delta</math>.]]
 
 
 
The spatial distribution of this steady streaming depends on the parameters of the problem, namely, 1) the ratio <math>\eta/\lambda</math> between the height <math>\eta</math> and the length <math>\lambda</math> of the bottom waviness, 2) the ratio <math>U_0/(\omega\lambda)</math> between the amplitude <math>U_0/\omega</math> of the fluid displacement oscillations and <math>\lambda</math>  (<math>U_0</math> and <math>\omega</math> denote the amplitude and angular frequency of the velocity oscillation of the fluid particles), 3) the ratio between <math>U_0/\omega</math> and the conventional thickness <math>\delta=\sqrt{2 \nu/\omega}</math> of the viscous bottom boundary layer (the reader should notice that <math>U_0/(\omega \delta)=U_0\delta/(2\nu)=R_\delta/2</math>, <math>R_\delta</math> being the Reynolds number characteristic of the bottom boundary layer). As these parameters are varied, different balances take place into momentum equation among the local acceleration, the convective acceleration, the pressure gradient and the viscous terms. A discussion of the relative importance of these terms and of the different approches which are used to determine the flow field as function of the values of the parameters (see among others Lyne <ref>Lyne W.H. (1971). Unsteady viscous flow over a wavy wall. J. Fluid Mech. 50, 33-48.</ref>, Sleath <ref name=S76>Sleath J.F.A. (1976). On rolling-grain ripples. J. Hydraul. Res. 14, 69-81.</ref>, Blondeaux <ref name=B90>Blondeaux P. (1990). Sand ripples under sea waves. Part 1. Ripple formation. J. Fluid Mech. 218, 1-17.</ref>, Vittori <ref name=V89>Vittori G. (1989). Non-linear viscous oscillatory flow over a small amplitude wavy wall J. Hydraulic Research 27 (2), 267-280.</ref>) can be found in Hara and Mei <ref name=HM>HaraT. and Mei C.C. (1990). Oscillating flows over periodic ripples. J. Fluid Mech. 211,
 
183-209.</ref>.
 
 
 
Figure 2 shows an example of the steady recirculating cells determined on the basis of Blondeaux's <ref name=B90></ref> analysis. When the steady velocity component close to the bed is directed from the troughs towards the crests of the bottom undulation and is strong enough to drag the sediments, the sediments tend to move from the troughs towards the crests. The tendency of sediments to pile up near the crests is opposed by the component of the gravity force acting in the down-slope direction. It follows that the growth of the amplitude of the bottom waviness is controlled by a balance between these two effects. If gravity prevails over drag, the amplitude decays, otherwise it grows leading to the appearance of ripples. Moreover, Sleath <ref name=S76></ref> argued that the effect is stronger for ripple wavelengths of the same order of magnitude as the amplitude of the fluid displacement oscillations since in this case particle settling locations will tend, after several cycles, to the nearest ripple crest.
 
Once formed, ripples do not continue to grow indefinitely because the steady streaming is modified by nonlinear effects and, as the ripples get steeper, an equilibrium configuration is attained.
 
 
 
 
 
==Models of ripple formation==
 
 
 
Idealized models allow to predict the conditions leading to the appearance of ripples and some of their geometrical characteristics. Even though they are not straightforward to be used, it is worthwhile to introduce them and briefly summarize the results they provide.
 
First of all, they show that the ripple wavelength and height do not depend on just one dimensionless parameter (this finding is also supported by some laboratory observations) and they suggest the most appropriate parameters to be used to improve the empirical predictors described in article [[Wave ripples]].
 
 
 
 
 
===Two-dimensional ripples===
 
 
 
The formation of two-dimensional ripples can be investigated by determining the time development of a two-dimensional random perturbation of the sea bottom characterized by a small amplitude and subject to a harmonic oscillatory flow (see [[Stability models]]). The assumption of a small amplitude of the perturbation allows the hydrodynamic and morphodynamic problems to be linearized and solved by considering the time development of a generic sinusoidal component of the bottom waviness characterized by a wavenumber <math>\alpha=2 \pi /\lambda</math> with <math>\lambda</math> of order <math>\delta</math>.
 
 
 
<math>y= \eta (x,t) = A(t) e^{i \alpha x} + c.c. . \qquad(1)</math>
 
 
 
To define the bottom profile (1), a Cartesian coordinate system is introduced with the origin located on the average bottom, the <math>y</math>-axis vertical and pointing upwards and the <math>x</math>-axis aligned in the direction of the fluid oscillations.
 
 
 
The hydrodynamic problem (the determination of the oscillatory flow over a wavy bed) was first solved by Sleath <ref name=S76></ref>, who considered moderate values of the Reynolds number such that the flow regime is laminar and both small and large values of the ratio between the ripple wavelength and the fluid displacement oscillations. Later Blondeaux <ref name=B90></ref> solved the problem for fluid displacement oscillations of the same order of magnitude as the wavelength of the bottom waviness.
 
 
 
The time development of the bottom waviness can be evaluated by considering sediment continuity (Exner) equation, after the introduction of an appropriate sediment transport predictor which relates the sediment flux to the bottom shear stress.
 
The linearized sediment continuity equation leads to
 
 
 
<math>\frac{d A}{dt}= \gamma (t) A(t) , \qquad(2) </math>
 
 
 
where <math>\gamma</math> is a complex quantity (<math>\gamma=\gamma_r + i \gamma_i</math>) (see Blondeaux<ref name=B90></ref>).
 
The time average <math>\overline \gamma_r</math> of <math>\gamma_r</math> describes the growth/decay of the bottom waviness, depending on its positive/negative value and is named 'growth rate', while the time average <math>\overline \gamma_i</math> of <math>\gamma_i</math> is related to the migration speed of the bottom forms. The periodic parts of <math>\gamma</math>, characterized by a vanishing time average (namely <math>\gamma_r-\overline \gamma_r</math> and <math>\gamma_i-\overline \gamma_i</math>), turn out to be small and describe the vertical and horizontal oscillations of the bottom profile, around its average position, which take place during the wave cycle.
 
If the small oscillations of the bottom profile around its averaged position are neglected, the bottom profile turns out to be described by
 
 
 
<math>\eta( x, t) = A_0 \exp \left[ \overline \gamma_r \tau \right]  \exp  \left[ i \alpha \left( x + \frac{ \overline \gamma_i}{\alpha} \tau \right) \right] + c.c. , \qquad (3) </math>
 
 
 
where <math>\tau=Q t</math> is the morphodynamic temporal scale and the value of <math>Q</math> depends on the formula used to quantify the sediment transport rate.
 
 
 
The analysis of Blondeaux <ref name=B90></ref> shows that, at the leading order of approximation, <math>\overline \gamma_i</math> vanishes and the bottom forms do not migrate, because of the symmetry of the problem. On the other hand the growth rate <math>\overline \gamma_r</math> of the bottom perturbation depends on the dimensionless parameters <math>R_\delta=U_0 \delta/\nu, R_p=\sqrt{\left( \rho_s/\rho-1\right) g d^3}/\nu, \psi_d=U_0^2/\left( \left( \rho_s/\rho-1\right) g d \right), s = \rho_s/\rho</math> or their combinations, where <math>\rho_s</math> and <math>\rho</math> are the densities of the sediment particles and the fluid, respectively, <math>g</math> is gravity acceleration and <math>d</math> is the grain size.
 
When <math>\overline{\gamma_r}</math> is plotted as function of <math>\alpha</math> for assigned values of the relative density <math>s</math> and the flow and sediment Reynolds numbers, denoted by <math>R_\delta</math> and <math>R_p</math> respectively, the results of Blondeaux<ref name=B90></ref> show that a critical value <math>\psi_{d,crit}</math> of <math>\psi_d</math> exists such that for values of <math>\psi_d</math> larger than the critical value, perturbation components characterized by wavenumbers falling within a restricted range have a positive growth rate and grow exponentially in time. Increasing values of <math>\psi_d</math> lead to an increase of the range of unstable wavenumbers while decreasing values of <math>\psi_d</math> make the unstable wavenumbers to collapse around a single value of the wavenumber (<math>\alpha_{crit}</math>) named 'critical wavenumber'.
 
 
 
[[Image: WaveRippleFormationFig2.jpg|thumb|400px|right|Figure 3. Region of existence of ripples and flat beds in the <math>(\psi_d,R_\delta)</math>-plane. Experimental observations for <math>5<R_d<15</math> and <math>s=2.65, \mu=0.15, n=0.4</math> (adapted from Blondeaux<ref name=B90></ref>).]]
 
 
 
Figure 3 shows the regions of existence of ripples (<math>\psi_d > \psi_{d,crit}</math>) and flat beds (<math>\psi_d < \psi_{d,crit}</math>) in the <math>(R_\delta,\psi_d)</math>-plane, as predicted by the stability analysis, along with the experimental observations of Blondeaux et al.<ref name=B88>Blondeaux P., Sleath J.F.A. and Vittori G. (1988). Experimental data on sand ripples in an oscillatory flow. Rep. 01/88. Inst. of Hydraulics, University of Genoa.</ref> who observed ripple formation using an oscillating tray apparatus.
 
 
 
[[Image: WaveRippleFormationFig3.jpg|thumb|400px|right|Figure 4. Critical value <math>\alpha_{crit}</math> of <math>\alpha</math> plotted versus the flow Reynolds number <math>R_\delta</math> for <math>s=2.65, \mu=0.15, n=0.4</math> and different values of <math>R_d=R_p\sqrt{\psi_d}</math>.]]
 
 
 
As pointed out in the accompanying article [[Wave ripples]], the dynamics of sediment grains and bed forms in oscillating trays is different from that forced by oscillatory flows in water tunnels or under sea waves. Indeed, although the fluid velocity distribution relative to axes fixed in the bed is the same, the forces acting on a sediment particle are not the same. This is because the force <math>\rho U_0 \omega V</math> on a particle of volume <math>V</math> due to the pressure gradient in an oscillatory flow is not equal to the inertia force <math>\rho_s U_0 \omega V</math> on a similar particle in an oscillating tray. On the other hand, the force on the particle due to the shear stress is the same in both cases. However, the measurements of Zala Flores and Sleath <ref name=ZS>Zala Flores N. and Sleath J.F.A. (1998). Mobile layer in oscillatory sheet flow. J. Geophys. Res. 103 (N. C6) 12783-12793.</ref> show that the movement of sediment grains is dominated by the shear stress and inertia plays a minor role when the parameter <math>\frac{U_0 \omega}{(s - 1) g} < 0.1</math> . The reader can easily verify that the data described in the following satisfy this condition. In figure 3, the theoretical values are obtained for <math>s=2.65</math> and different values of <math>R_p</math> such that <math>R_d=U_0d/\nu=R_p \sqrt{\psi_d}=10</math>, while the experimental observations are characterized by values of <math>R_d</math> falling in the range <math>(5,15)</math>.
 
 
 
Notwithstanding the quantitative differences between the theoretical results and the experimental data, the stability analysis appears to provide a reliable description of the process which leads to ripple formation. Moreover, the model shows that coarser sediments, which are characterized by larger values of <math>R_p</math>, give rise to longer ripples. This theoretical finding is in agreement with laboratory measurements as shown in figure 4, where the critical values of <math>\alpha_{crit}</math> are plotted versus the Reynolds number <math>R_\delta</math> of the bottom boundary layer for different values of the sediment Reynolds number <math>R_d=R_p \sqrt{\psi_d}</math>.
 
 
 
The comparison between the theoretical values and the experimental measurements shows that the analysis underpredicts the observed wavelengths but a qualitative agreement with the laboratory measurements can be observed. Moreover, figure 3 shows that the ripple wavelength can not be predicted on the basis of just one parameter since both <math>R_\delta</math> and <math>R_d</math> affect the length of the bedforms.
 
 
 
 
 
==Rolling-grain ripples and vortex ripples==
 
 
 
[[Image: WaveRippleFormationFig4.jpg|thumb|350px|left|Figure 5. Visualization of the sediments picked-up from the bed and carried into suspension by the vortices shed at the crests of large amplitude ripples in an oscillatory flow (courtesy of Dr. Megale).]]
 
 
 
A linear approach, which considers ripples characterized by small (strictly infinitesimal) amplitudes, cannot follow the time development of the bottom forms for long times and determine their equilibrium amplitude. The prediction of the equilibrium amplitude of ripples is quite important since for large amplitudes the flow separates from the crests of the ripples and vortices are shed which modify the mechanism  of sediment transport (see figure 5).
 
 
 
A model to determine the temporal growth of ripples for long times and to predict their equilibrium amplitude was developed by Vittori and Blondeaux<ref name=VB90>Vittori G., Blondeaux P. (1990). Sand ripples under sea waves. Part 2. Finite amplitude development. J. Fluid Mech. 218, 19-39.</ref> taking into account nonlinear effets but assuming that they are weak, i.e. when the amplitude of the ripples is moderate and the bottom boundary layer does not separate from the ripple crests.
 
 
 
[[Image: WaveRippleFormationFig4a.jpg|thumb|450px|left|Figure 6a. Rolling-grain ripples (courtesy of John F.A. Sleath).]]
 
[[Image: WaveRippleFormationFig4b.jpg|thumb|450px|right|Figure 6b. Vortex ripples (courtesy of John F.A. Sleath).]]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Vittori and Blondeaux<ref name=VB90></ref> showed that, for fixed values of <math>s</math> and <math>R_p</math>, the plane <math>(R_\delta,\psi_d)</math> can be divided into three regions.
 
In the first region, defined by values of <math>\psi_d</math> smaller than <math>\psi_{d,crit}</math>, the flat bottom is stable and ripple do not appear. For values of <math>\psi_d</math> larger than <math>\psi_{d,crit}</math> the linear analysis of Blondeaux<ref name=B90></ref> predicts the formation of ripples. In this region, Vittori and Blondeaux<ref name=VB90></ref> identified two sub-regions. In one sub-region the ratio between the predicted height and length of the ripples at equilibrium is smaller than about <math>0.1</math> and the analysis, according with the criterion suggested by Sleath<ref name=S84>Sleath J.F.A. (1984) Sea bed mechanics. John Wiley and Sons.</ref>, predicts the appearance of rolling grain ripples as equilibrium bedforms (see figure 6a). In the other sub-region, the analysis shows that no equilibrium is possible assuming that nonlinear effects are weak. In this case the amplitude of the bottom forms grows and nonlinear effects become increasingly more important till the boundary layer separates from the ripple crests and vortex ripples are generated (see figure 6b).
 
 
 
A comparison between theoretical predictions and the experimental observations of Blondeaux et al. <ref name=B88></ref> and Horikawa and Watanabe<ref name=HW></ref> is shown in figure 7.
 
 
 
[[Image: WaveRippleFormationFig5ab.jpg|thumb|800px|center|Figure 7. Regions in the <math>(R_\delta,\psi_d)</math>-plane where a flat bed, rolling-grain ripples, and vortex ripples are expected to appear according to Vittori and Blondeaux<ref name=VB90></ref>. Comparison between the theoretical predictions and the laboratory observations. Left panel: <math>R_d=R_p\sqrt{\psi_d}=10, \beta=0.15, s=2.65</math> and experimental data by Blondeaux et al. <ref name=B88></ref> (<math>5<R_d<10</math>); Right panel: <math>R_d=R_p\sqrt{\psi_d}=40, \beta=0.15, s=2.65</math> and experimental data by Horikawa and Watanabe <ref name=HW>Horikawa K. and Watanabe, A. (1968). Laboratory study on oscillatory boundary layer flow. Proc. 11th Coastal Eng. Conf., 467-486.</ref> (<math>30<R_d<50</math>).]]
 
 
 
 
 
 
 
===Brick-pattern ripples===
 
 
 
[[Image: WaveRippleFormationFig6a.jpg|thumb|350px|right|Figure 8. Brick-pattern ripples (courtesy of John F.A. Sleath).]]
 
 
 
Vittori and Blondeaux<ref name=VB92>Vittori G., Blondeaux P. (1992). Sand ripples under sea waves. Part 3. Brick-pattern ripple formation. J. Fluid Mech. 239, 23-45.</ref> developed an idealized model for the formation of brick-pattern ripples (see figure 8) by considering the time development of a sandy bottom subject to an oscillatory flow when three-dimensional initial disturbances of the bottom profile are present. The analysis shows that brick-pattern ripples can be originated by the simultaneous growth of two-dimensional and three-dimensional components of the initial disturbance, which interact with a mechanism similar to that described by Craik<ref name=C>Craik A.D.D. (1971). Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393-413.</ref> in a different context. The results allow to identify the regions in the parameter space where brick-pattern ripples appear.
 
 
 
In summary, on the basis of the analyses of Blondeaux<ref name=B90></ref>, Vittori and Blondeaux<ref name=VB90></ref> and Vittori and Blondeaux<ref name=VB92></ref> the plane <math>(R_\delta, \psi_d)</math> (for fixed values of <math>s</math> and <math>R_d</math>) can be divided into four different regions where the stability analyses predict i) stable flat bed, ii) rolling-grain ripples, iii) vortex ripples, iv) brick-patterns ripples. Figure 9 shows the different regions in the <math>(R_\delta,\psi_d)</math>-plane for fixed values of <math>s</math> and <math>R_d</math> along with the experimental results of Sleath and Ellis<ref name=SE>Sleath J.F.A. and Ellis A.C. (1978). Ripple geometry in oscillatory flow. Univ. Cambridge Dept. Engr. Rept. A/Hydraulics TR2, 15 pp.</ref> and Horikawa and Watanabe<ref name=HW></ref>.
 
 
 
[[Image: WaveRippleFormationFig7.jpg|thumb|400px|right|Figure 9. Limiting curves dividing the <math>(R_\delta, \psi_d)</math>-plane in regions where a flat bed, rolling grain ripples, two-dimensional vortex ripples, brick-pattern ripples are expected to form (<math>R_d=R_p\sqrt{\psi_d}=40, s=2.65, \beta= 0.15</math>). Also shown are experimental data by Sleath and Ellis<ref name=SE></ref> and Horikawa and Watanabe<ref name=HW></ref> for <math>35 < R_\delta < 45</math> (white points = rolling grain ripples, black points = vortex ripples, triangles = brick pattern ripples). Adapted from Vittori and Blondeaux<ref name=VB92></ref>.]]
 
 
 
 
 
 
 
 
 
 
 
 
 
===Tile ripples===
 
 
 
Field and experimental observations show the existence of hexagonal ripples, named tile ripples
 
(see article [[Wave ripples]]). Roos and Blondeaux<ref name=RB>Roos P.C., Blondeaux P. (2001). Sand ripples under sea waves. Part 4. Tile ripple formation. J. Fluid Mech. 447, 227-246.</ref> developed a model similar to that of Vittori and Blondeaux<ref name=VB92></ref> but they considered the forcing flow generated by the simultaneous presence of two surface waves characterized by the same angular frequency but different amplitudes and directions of propagation, such as the wave field which can be observed when a wave is partially reflected by a coastal structure. In this case, close to the bottom, the irrotational flow is not unidirectional but characterized by an elliptical behaviour.
 
The model of Roos and Blondeaux <ref name=RB></ref> predicts the conditions for the formation of tile ripples.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
==Models of ripple migration due to waves only==
 
 
 
When a steady current is superimposed to surface waves, the so-called wave-current ripples are observed, the characteristics of which are somewhat between those of wave ripples, previously described, and those of current ripples. Since the flow, generated close to the bottom by sea waves, is characterized by an oscillatory velocity component and by a steady velocity component originated by nonlinear effects <ref>Longuet-Higgins M.S. (1953). Mass transport in water waves. Philos. Trans. R. Soc. 345. 535-581.</ref>, the ripples generated by sea waves of large amplitude have geometrical and kinematical characteristics which are similar to those of the ripples generated by the interaction between waves and currents.
 
 
 
[[Image: WaveRippleFormationFig9.jpg|thumb|500px|right|Figure 10. Equilibrium profile of the ripples predicted by the stability analysis of Blondeaux et al. <ref name=B16>Blondeaux P., Foti E., Vittori G. (2015). A theoretical model of asymmetry wave ripples. Phil. Trans. R. Soc. A 373 (2033). pii: 20140112. doi: 10.1098/rsta.2014.0112.</ref> for <math>R_\delta=15, R_d= 25, \frac{\sqrt(\psi_d-\sqrt{\psi_{dc}}}{\sqrt{\psi_{dc}}} = 0.1, 2 \pi \delta/\lambda= 0.364, \delta/L = 0.004</math> (continuous line, considering <math>O(\delta/L)</math>-effects; broken line, neglecting <math>O(\delta/L)</math>-effects). The ratio between the ripple height <math>\eta</math> and the ripple wavelength <math>\lambda</math> is abount <math>0.067</math>.]]
 
 
 
The distinguishing geometric characteristic of the ripples generated by large amplitude waves
 
is the asymmetry of their profile (see figure 10).
 
 
 
Moreover, the presence of a steady streaming induces the migration of the bottom forms. Blondeaux et al. <ref name=B20>Blondeaux P., Foti E., Vittori G. (2000). Migrating sea ripples. European Journal of Mechanics - B/Fluids 19 (2), 285-301.</ref> investigated whether the steady velocity component has a destabilizing or a stabilizing effect on the formation of ripples. Moreover, they determined the migration speed of the ripples.
 
This information is of practical interest since it is common practice to evaluate the average sediment transport rate from measurements of ripple migration assuming that the sediment transport rate is related to the migration speed times the height of the ripples  <ref name=FD>Fredsøe J. and Deigaard R. (1992). Mechanics of coastal sediment transport. Advances Series on Ocean Engineering 3 World Scientific.</ref>.
 
 
 
When nonlinear effects due to large amplitude sea waves are taken into account to determine the forcing flow, the value of <math>\gamma</math> turns out to have both a real <math>\gamma_r</math> part and imaginary <math>\gamma_i</math> part. The value of <math>\overline \gamma_r</math> differs from that determined by Blondeaux<ref name=B90></ref> because of terms of order <math>a/L</math> and the migration speed of the bottom forms is proportional to the steepness of the surface wave (<math>a</math> and <math>L</math> denote the amplitude and the length of the sea waves). A discussion of the results of the linear analysis can be found in Blondeaux et al. <ref name=B20></ref>, where the interested reader can also find an exhaustive discussion of the influence of second order terms in the wave slope on the stability of the sea bottom.
 
 
 
Figure 11 shows a comparison between the migration speed predicted by the theoretical analysis
 
and that measured by Blondeaux et al. <ref name=B20></ref> during laboratory experiments.
 
 
 
[[Image: WaveRippleFormationFig10.jpg|thumb|450px|left|Figure 11. Theoretical and experimental values of the dimensioless migration speed of ripples plotted versus <math>R_\delta</math>. The theoretical values are obtained by evaluating the sediment transport rate by means of Hallermeier's formula <ref>Hallermeier R.J. (1982). Oscillatory bedload transport: data review and simple formulation. Continental Shelf Res. 1, 159-190.</ref> and the migration speed is scaled introducing the morphodynamic time scale <math>\tau</math>  <ref name=B20></ref>. The experimental measurements are for <math>5<R_d<10</math> (white circles), <math>10<R_d < 15</math> (black circles), <math>15 <R_d < 20</math> (white triangles), <math>20 < R_d< 25</math> (black triangles), <math>25 < R_d < 30 </math> (while diamond).]]
 
 
 
[[Image: WaveRippleFormationFig11.jpg|thumb|450px|right|Figure 12. Theoretical value of the symmetry index plotted versus <math>U_s/U_0</math>, where the value of <math>U_s/U_0</math> for a monochromatic wave is assumed to be <math>3 \pi \delta R_\delta/(4L) </math>. <math>R_d = 25</math> and <math>R_\delta = 15,30</math> are values characteristic of Blondeaux et al.'s<ref></ref> (2000) experiments. The experimental data of  Inman<ref name=I></ref>, Tanner<ref name=T></ref>, Blondeaux et al. <ref name=B20></ref> are also shown. The values of <math>\psi_d</math> are related to <math>F_d</math> by <math>\psi_d=F_d^2</math>.]]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
  
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'''Resilience and resistance'''
  
  
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{{Definition|title=Resistance
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|definition= The capacity to weather a disturbance without loss (Lake 2013<ref name=L>Lake, P.S. 2013. Resistance, Resilience and Restoration. Ecological Management and Restoration 14: 20-24</ref>). }}
  
  
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{{Definition|title=Resilience
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|definition=(1) the capability to anticipate, prepare for, respond to, and recover from significant multihazard threats with minimum damage to social well-being, the economy, and the environment (sometimes called 'socio-ecological resilience')(Olsen et al. 2019<ref name=O>Olsson, S., Melvin, A. and Giles, S. (eds.) 2019. Climate change and ecosystems. Procs. Sackler Forum on Climate Change and Ecosystems, Washington, DC, November 8-9, 2018, organized by the National Academy of Sciences and The Royal Society</ref>);
  
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(2) the capability of a (socio-)ecological system to remain within a stability domain when subjected to environmental change, while continually changing and adapting yet remaining within critical thresholds (sometimes called 'general resilience') (Folke et al. 2010<ref name=F>Folke, C., Carpenter, S. R., Walker, B., Scheffer, M., Chapin, T. and Rockstrom, J. 2010. Resilience thinking: integrating resilience, adaptability and transformability. Ecology and Society 15(4): 20</ref>; Scheffer 2009<ref>Scheffer, M. 2009. Critical transitions in nature and society. Princeton University Press, Princeton, New Jersey, USA</ref>; Brand and Jax 2007<ref name=BJ>Brand, F.S. and K. Jax. 2007. Focusing the meaning(s) of resilience: resilience as a descriptive concept and a boundary object. Ecology and Society 12(1):23</ref>);
  
 +
(3) the capacity to experience shocks while retaining essentially the same function, structure, feedbacks, and therefore identity (sometimes called 'ecological resilience') (Brand and Jax 2007<ref name=BJ/>; DEFRA 2019<ref name=DEFRA>Haines‐Young, R. and Potschin. M. (eds.) 2010. The Resilience of Ecosystems to Environmental Change (RECCE). Overview Report, 27 pp. Defra Project Code: NR0134</ref>), which is closely related to the concept of 'ecosystem resistance': the amount of disturbance that a system can withstand before it shifts into a new regime or an alternative stable state (Holling 1973<ref>Holling, C.S. 1973. Resilience and stability of ecological systems. Annual Rev. Ecol. Syst. 4: 1–23. doi: 10.1146/annurev.es.04.110173.000245</ref>; Gunderson 2000<ref>Gunderson, L.H. 2000. Ecological Resilience - in Theory and Application. Annual Review of Ecology and Systematics 31:425-439.</ref>);
  
 +
(4) the capacity of an ecosystem to regain its fundamental structure, processes, and functioning (or remain largely unchanged) despite stresses, disturbances, or invasive species (e.g., Hirota et al., 2011<ref>Hirota,M., Holmgren,M., Van Nes, E. H, and Scheffer,M. 2011. Global resilience of tropical forest and savanna to critical transitions. Science 334: 232–235. doi: 10.1126/science.1210657</ref>; Chambers et al., 2014<ref>Chambers, J. C., Bradley, B. A., Brown, C. S., D’Antonio, C., Germino, M. J., Grace, J. B., et al. 2014. Resilience to stress and disturbance, and resistance to Bromus tectorum L. invasion in the cold desert shrublands of western North America. Ecosystems 7: 360–375. doi: 10.1007/s10021-013-9725-5</ref>; Pope et al., 2014<ref>Pope, K. L., Allen, C. R., and Angeler, D. G. 2014. Fishing for resilience. T. N. Am. Fisheries Soc. 143: 467–478. doi: 10.1080/00028487.2014.880735</ref>; Seidl et al., 2016<ref>Seidl, R., Spies, T. A., Peterson, D. L., Stephens, S. L., and Hick, J. A. 2016. Searching for resilience: addressing the impacts of changing disturbance regimes on forest ecosystem services. J. Appl. Ecol. 53 : 120–129. doi: 10.1111/1365-2664.12511</ref>), which can be measured by the time needed to recover its original state (sometimes called 'engineering resilience'<ref name=L>Lake, P.S. 2013. Resistance, Resilience and Restoration. Ecological Management and Restoration 14: 20-24</ref>).
 +
}}
  
  
  
 +
==Introduction==
 +
Coastal and marine ecosystems are affected by environmental disturbance at a variety of spatio-temporal scales. The organisms inhabiting these systems are adapted to such disturbance, either by being tolerant of these conditions or by playing a role in one or more of the successional stages that follow during ecosystem recovery.
  
 +
If all species in the system were tolerant to a particular perturbation, very little would change at the ecosystem level, and we could call the system resistant to this disturbance. However, often a disturbance, such as a temporary very low oxygen level, affects a substantial proportion of the organisms dramatically, either causing them to die, or forcing them to rapidly migrate to more favorable parts of the environment. Such an adverse disturbance could locally defaunate a certain volume in the pelagic or a certain area of hard or soft substrate. Such destruction at a local scale does not mean the end of local functioning. Usually organisms are available at a larger spatial scale that can re-colonize the affected area, according to their particular tolerances and abilities to favorably affect their local environment.
  
 +
The term resilience has been defined in different ways, illustrated in the definition above. According to DEFRA (2019<ref name=DEFRA/>) there is limited consensus in the literature about how resilience can be characterized and assessed. The term resilience is sometimes used to represent some kind of normative proposition about what kinds of ecosystem characteristics are desirable or necessary in the context of sustainable development, reflecting particular cultural and philosophical assumptions<ref name=DEFRA/>. However, the resistance of an ecosystem (see the definition above) to changing conditions and the rate of recovery following some disruptive event are generally considered major components of resilience that can in principle be expressed in quantitative terms.
  
 +
Other attributes such as the capacity of ecosystems to transform and adapt in the face of environmental change (i.e. system's ability to re-organize itself) are more difficult to translate to practice. According to Dawson et al. (2010<ref name=D>Dawson, T.P., Rounsevell, M.D.A., Kluvankova‐Oravska, T., Chobotova V. and Stirling, A. 2010. Dynamic properties of complex adaptive ecosystems: implications for the sustainability of services provision. Biodiversity and Conservation 19: 2843‐2853</ref>), resilience concerns the response of ecosystems to changing environmental conditions and must be looked at alongside other additional dynamic features, namely durability, robustness and stability. These concepts can be defined as<ref name=D/>:
 +
* Durability:  ability to cope with a chronic stress, but the source of this stress is endogenous;
 +
* Robustness: ability to recover or maintain the systems' social-ecological functions in the face of an external and chronic driver;
 +
* Stability:  system’s tolerance to transient and endogenous shocks or disruptions.
  
The analysis of Blondeaux et al. <ref name=B20></ref> neglects the nonlinear effects related to the amplitude of the bottom forms. Hence, the bottom profile turns out to be sinusoidal and its amplitude can not be determined. To describe the process, which shapes the bottom profile giving rise to bottom forms characterized by crests sharper than the troughs and up-current slopes smaller than the down-current slopes, Blondeaux et al. <ref name=B16></ref> considered the interaction of the bottom perturbation with itself.
+
Both resistance and resilience cause an ecosystem to remain relatively unchanged when confronted to a disturbance, but in the case of resistance no internal re-organization and successional change is involved. In contrast, resilience implies that the system is internally re-organizing, perhaps through a mozaic of patches that are at different stages of re-assembly. System responses to changing environmental conditions are displayed schematically in Fig. 1, corresponding to different resilience characteristics.
  
Figure 10 shows the equilibrium profile of a ripple obtained by means of the theory, for fixed values of the parameters along with that obtained neglecting <math>O(a/L)</math> effects. The characteristic profile of ripples affected by a steady streaming can be recognized from the lee side of the ripple being steeper than the stoss side. The steady velocity component in figure 10 is from the right to the left. In this case the ratio <math>\lambda_2/\lambda_1</math> turns out to be <math>1.81</math>. The experimental data by Inman<ref name=I>Inman D.L. (1957). Tech. Mem. U.S. Beach Erosion N. 100.</ref>, Tanner<ref name=T>Tanner W.F. (1971). Numerical estimates of ancient waves, water depth and fetch. Sedimentology 16, 71-88.</ref>, who used sand, and those described in Blondeaux et al. <ref name=B20></ref>, who used high density plastic particles, show that the symmetry index increases as the ratio between the steady velocity component and the amplitude of the oscillating one increases but a limiting value exists. The same dependence is observed in the theoretical results presented in figure 12. Interestingly, for strong steady currents, the different curves tend to converge towards a common value that appears to be the maximum value of the symmetry index and ranges between <math>2</math> and <math>3</math>.
+
[[Image:ResilienceTrajectories.jpg|thumb|900px|center|Figure 1. Schematic representation of the trajectories of a (socio-)ecological system in a plane defined by the system state (fundamental structure, processes, and functioning - vertical axis) and the change of environmental conditions (horizontal axis), for different resilience characteristics (a, b, c, d). The initial state corresponds to the position on the graph at the vertical axis (zero change in environmental conditions). In all situations the ecosystem is assumed to collapse irreversibly (down to the horizontal axis) when the change in environmental conditions is much greater than the systems' resistance. The angle <math>\alpha</math> represents the rate at which the system recovers when the change in environmental conditions is reduced (small <math>\alpha</math> means slow recovery, large <math>\alpha</math> means fast recovery). Panel a: Resilience characterized by high resistance (definition 3) and slow recovery (definition 4). Panel b: Resilience characterized by low resistance and fast recovery. Panel c: Resilience characterized by a shift to an alternative stable system state. Panel d: Low resilience, characterized by low resistance and slow recovery.]]
A quantitative comparison between experimental observations and theoretical findings can be made looking at table 1, which shows the values of <math>\lambda_2/\lambda_1</math> detected during experiments nr. 39, 40, 42 of Blondeaux et al. <ref name=B20></ref> along with the theoretical values computed for the same parameters of the problem.
 
  
{|  style="border-collapse:collapse;background:ivory;" cellpadding=5px align=center width=50%
 
|+ Table 1. Experimental and theoretical values of <math>\lambda_2/\lambda_1</math> for the experiments of Blondeaux et al. <ref name=B20></ref> characterized by the presence of rolling grain ripples.
 
|- style="font-weight:bold; text-align:center; background:lightgrey"
 
! width=10% style=" border:1px solid gray;"| exp. nr.
 
! width=10% style=" border:1px solid gray;"| <math>(\lambda_2/\lambda_1)_{exp.}</math>
 
! width=10% style=" border:1px solid gray;"| <math>(\lambda_2/\lambda_1)_{theor.}</math>
 
|-
 
| style="border:1px solid gray;"|
 
| style="border:1px solid gray;"|
 
| style="border:1px solid gray;"|
 
|-
 
| style="border:1px solid gray;"| 39
 
| style="border:1px solid gray;"| 1.19
 
| style="border:1px solid gray;"| 1.17
 
|-
 
| style="border:1px solid gray;"| 40
 
| style="border:1px solid gray;"| 1.23
 
| style="border:1px solid gray;"| 1.17
 
|-
 
| style="border:1px solid gray;"| 42
 
| style="border:1px solid gray;"| 1.30
 
| style="border:1px solid gray;"| 1.20
 
|}
 
  
 +
When considering the potential effect of a certain type of disturbance it is thus useful to ask two questions:
 +
# Will the species of this system be able to tolerate it (implying resistance), and if not,
 +
# Is recovery possible through a successional trajectory, back to the same, or at least a desirable, ecosystem state (implying resilience)?
 +
Resistance breaks down when uni-directional ongoing change acts faster than the organisms' ability to adapt their tolerances. If uni-directional ongoing change is this fast (even if gradual), the system will not be sufficiently resilient either, as full recovery through succession will then not be possible. Recovery from sudden and local disturbance is often possible through recolonization, but the rate of recovery will depend crucially on the spatial extent of disturbance. For example, recovery from anoxia could take 5 to 8 months at the scale of square meters (Rossi et al. 2009<ref name=R>Rossi, F., Vos, M. & Middelburg, J.J. 2009. Species identity, diversity and microbial carbon flow in reassembling macrobenthic communities. Oikos 118: 503-512.</ref>), but could take 5 to 8 years at the scale of a whole bay (Diaz & Rosenberg 1995<ref>Diaz, R.J. & Rosenberg, R. 1995. Marine benthic hypoxia: a review of its ecological effects and the behavioural responses of benthic macrofauna. Oceanogr. Mar. Biol. Annu. Rev. 33:245-303.</ref>).
  
For experiments nr. 30, 35, 36, 41, the theory predicts the formation of vortex ripples and hence the symmetry index can not be computed.
+
According to definition (4), the speed at which an ecosystem returns to its former state following a (minor) disturbance can be considered a measure of resilience. The idea is that a system with a short return time is more resilient than one with a long return time. Such resilience measured as (1 / the return time to a stable equilibrium) has also been called ''engineering resilience''. It has however a long history of use among ecologists (Pimm 1982<ref>Pimm, S.L. 1982. Food Webs. The University of Chicago Press.</ref>, DeAngelis 1992<ref>DeAngelis, D.L. 1992. Dynamics of Nutrient Cycling and Food Webs. Chapman and Hall, London.</ref>, Vos et al. 2005<ref>Vos, M., Kooi, B.W., DeAngelis, D.L. & Mooij, W.M. 2005. Inducible defenses in food webs. In: Dynamic Food Webs. Multispecies Assemblages, Ecosystem Development and Environmental Change. Eds. P.C. de Ruiter, V. Wolters & J.C. Moore. Academic Press. Pp. 114-127.</ref>). Resilience is also used in a way that more closely resembles the definition of resistance. ''Ecological resilience'' was defined as the amount of disturbance that an ecosystem could withstand without changing self-organized processes and structures (definition 3).
  
 +
Resilience of coastal systems largely depends on biodiversity, which is a major requirement for allowing ecosystems to adapt to changing conditions. The human impact on the environment through pollution, fisheries, sediment erosion / deposition and global climate change has brought about much faster change than would occur under natural conditions, putting severe stress on many ecosystems. Without genetic diversity, natural selection cannot occur and if natural selection is limited, adaptation is impossible. Preservation of biodiversity and, more specifically, genetic diversity is therefore of paramount importance for successful adaptation to our rapidly changing environments. However, biodiversity may not always protect ecosystems from major abiotic disturbances (Folke et al. 2004<ref>Folke, C., Carpenter, S., Walker, B., Scheffer, M., Elmqvist, T., Gunderson, L. & Holling, C.S. 2004. Regime Shifts, Resilience, and Biodiversity in Ecosystem Management. Annual Review of Ecolog and Systematics 35:557-581.</ref>).
  
==Ripple formation: the turbulent boundary layer case==
+
==Resilience through recolonization==
  
[[Image: WaveRippleFormationFig12.jpg|thumb|500px|right|Figure 13. Ratio between the amplitude <math>U_0/\omega</math> of the fluid displacement oscillations and the wavelength <math>\lambda</math> of the ripples plotted versus the parameter <math>\rho d/(\rho_s-\rho) g T^2</math>. The continuous lines are the theoretical predictions of the linear stability anaysis of Foti and Blondeaux<ref name=FB></ref> developed assuming the flow regime to be turbulent. The experimental data of Manohar<ref>Manohar M. (1955) 'Mechanics of bottom sediment movement due to wave action', Tech. Memo. 75, 121 pp., U.S. Army Corps of Eng., Beach Erosion Board, Washington, D.C.</ref> (points) and Sleath<ref name=S76></ref> (open circles) are taken from the book of Sleath<ref></ref> (adapted from Foti and Blondeaux<ref name=FB></ref>).]]
+
To understand resilience of ecosystems it is essential to understand what drives succession within these ecosystems. Succession determines how, and how fast, communities return to their original state, or perhaps enter a new state. Many aspects of succession can be understood in terms of trade-offs between the ability to be either a good early (re)colonizer, or a good competitor. Succession involves a gradual replacement of colonizer/competitor species according to the degree to which they tolerate, facilitate or inhibit certain environmental conditions and other species (Rossi et al. 2009<ref name=R/>). The extent to which processes of (re)colonization and succession can take place largely determines the recovery of ecosystems after major disruption and is therefore an essential characteristic of the resilience of ecosystems.  
  
The analyses previously summarized explain the appearance of ripples and predict their characteristics at incipient formation but for moderate values of the Reynolds number, such that the flow regime is laminar. For field conditions, the Reynolds number <math>R_\delta</math> is often larger than its critical value and the flow regime is turbulent. The model of Blondeaux (1990) was extended to the turbulent regime by Foti and Blondeaux<ref name=FB>Foti E., Blondeaux P. (1995). Sea ripple formation: the turbulent boundary layer case. Coastal Eng., 25 (3-4), 227-236.</ref> who considered the Reynolds averaged momentum equations and used a simple turbulence model. They evaluated, in particular, the Reynolds stresses by using a constant turbulent eddy viscosity <math>\nu_T</math>. In order to obtain a reasonable velocity profile, as suggested by Sleath<ref name=S91>Sleath J.F.A. (1991). Velocities and shear stresses in wave-current flows. J. Geophys. Res. 96(C8), 15237-15244.</ref> and by Engelund and Fredsøe<ref name=EF>Engelund F. and Fredsøe J. (1982). Sediment ripples and dunes. Ann. Rev. Fluid Mech. 14, 13-37.</ref> in another context, they further replaced the no-slip condition at the bottom by a partial slip condition.
+
In this context, it is important to consider the spatial component of ecosystem resilience. Diversity of structurally and functionally connected landscapes, rich in resources and species, promotes the flow or movement of individuals, genes, and ecological processes. Below certain thresholds of connectivity the capacity to regain structure and function after perturbation is lost (Holl and Aide, 2011; Rudnick et al., 2012;McIntyre et al., 2014; Rappaport et al., 2015; Ricca et al., 2018). Chambers et al. (2019<ref name=CAC>Chambers, J.C., Allen, C.R. and Cushman, S.A. 2019. Operationalizing Ecological Resilience Concepts for Managing Species and Ecosystems at Risk. Front. Ecol. Evol. 7:241. doi: 10.3389/fevo.2019.00241</ref>), based on Allen et al. (2016<ref> Allen, C. R., Angeler, D. G., Cumming, G. S., Folk, C., Twidwell, D., and Uden, D. R. 2016. Quantifying spatial resilience. J. Appl. Ecol. 53, 625–635. doi: 10.1111/1365-2664.12634</ref>), have therefore introduced the concept of  'spatial resilience', which is a measure of how spatial attributes, processes, and feedbacks vary over space and time in response to disturbances and affect the resilience of ecosystems. Self-organization through strong feedbacks at multiple scales and high levels of functional diversity and redundancy, stabilizes the system with respect to disturbances within the range of historic variability.
Figure 13 shows a comparison between laboratory data and the theoretical predictions made by means of this model.
 
  
 +
When creating Marine Protected Areas, the sources of populations at all stages of succession should be protected, to preserve 'ecological memory' to the fullest possible extent. This includes protecting not only 'high quality' habitats that harbour healthy mature communities, but also 'low quality' and disturbed habitats that are required for those species that contribute to early recovery of perturbed areas (Rossi et al. 2009<ref name=R/>). The selection of Marine Protected Areas thus involves evaluating
 +
the number, size, and spatial configuration of habitat fragments and degree of connectivity required to support restoration of ecosystems and conservation of focal habitats and species<ref name=CAC/><ref name=O/>.
  
 +
==Resistance to changes in abiotic and biotic factors==
  
 +
Community composition and ecosystem function may change very little under environmental change when the organisms can adapt to such change or tolerate it for some time (when the change is only temporary). However, all organisms have bounds to what they can temporarily or permanently tolerate, and when change exceeds some of these limits, the community composition and ecosystem functioning is likely to change.
  
 +
It is unlikely that communities can be resistant to ongoing gradual change, such as global warming. Acclimation and phenotypic plasticity do not suffice to maintain the system as it is. Genetic adaptation could allow community members to track such abiotic environmental change, but it is more likely that the area where the community is functioning will be invaded by species that function well at higher temperatures. The original species will thus have to deal with new competitors and predators, in addition to the changed abiotic factor. To some extent the original community can track the preferred temperature range, by moving spatially to greater depths or to alternative geographic areas. But these new areas are likely to differ in other ecological aspects such as water pressure, light climate and perhaps speeds of water flow etc.
  
 +
==Adaptation and the consequences of mortality at different trophic levels==
  
 +
External disturbance interacts with internal mechanisms that shape community structure. To understand how an increased mortality of top-predators will affect the entire food chain, it is essential to understand how processes of mutual adaptation within food chains already give shape to existing patterns such as trophic structure (how biomass in ecosystems is partitioned between trophic levels).
  
 
+
Abundances at different trophic levels (such as algae, herbivores, carnivores and top-predators) and their responses to increased mortality (as under environmental change) depend critically on different mechanisms of adaptation within food chains and on the importance of population density at each of these trophic levels. However, different types of adaptation to living in a food chain context (balancing the need to acquire resources with the need to avoid predation) can often have similar consequences. For example, micro-evolution of behaviour, species replacement and induced defenses at a middle trophic level may all have similar effects on trophic level abundances in disturbed food chains (Abrams and Vos 2003<ref>Abrams, P.A & Vos, M. 2003. Adaptation, density dependence and the responses of trophic level abundances to mortality. Evolutionary Ecology Research 5: 1113-1132</ref>).
 
 
 
 
 
 
 
 
==Vortex ripples==
 
 
 
[[Image: WaveRippleFormationFig13.jpg|thumb|500px|right|Figure 14. Vorticity contours of the oscillatory flow over a wavy wall <math>R_\delta=50, \eta/\lambda=0.15, U_0/(\omega \lambda)=0.75</math> (The vorticity isolines are <math>0.15 U_0/\omega</math> apart, thick lines = clockwise vorticity, thin lines = counter-clockwise vorticity). Panel a) <math>\omega t = \pi/2</math>; Panel b) <math>\omega t = \pi</math>; Panel c) <math>\omega t =5 \pi/4</math>;  Panel d) <math>\omega t = 3 \pi/2</math>; Panel e) <math>\omega t = 2 \pi</math>. Adapted from Blondeaux and Vittori<ref name=BV91></ref>.]]
 
 
 
The models based on linear and weakly nonlinear stability analyses are no longer valid when the parameters of the problem are far from the critical conditions.
 
In this case, nonlinear effects become relevant and a perturbation approach can be used no longer. Under these circumstances, only numerical simulations of momentum and Exner equations can be used to determine the fluid flow and the time development of the bottom forms.
 
 
 
The oscillatory flow over vortex ripples was first determined by the numerical integration of momentum equation by Shum<ref>Shum K.T. (1988) A numerical study of vortex dynamics over rigid ripples. PhD Thesis M.I.T. Dep. of Civil Engineering, Cambridge, Mass.</ref> and Blondeaux and Vittori<ref name=BV91>Blondeaux P., Vittori G. (1991). Vorticity dynamics in an oscillatory flow over a rippled bed. J. Fluid. Mech. 226, 257-289.</ref>.
 
Figure 14 shows the spanwise component of vorticity over two-dimensional vortex ripples
 
at different phases from the beginning of the numerical simulation, as computed by the latter authors. When the fluid moves from the left to the right, clockwise vorticity is generated along the bottom profile and tends to roll up and to generate a well defined vortex (figure 14a,b).
 
 
 
As the external fluid velocity reverses its direction, the clockwise vortex is no longer fed but it is convected from the right to the left by the external flow (figure 14c,d). Simultaneously, counter-clockwise vorticity is shed from the crest and the phenomenon repeats similarly during the following half cycle (figure 14e).
 
Of course, the size, the strength and the number of vortex structures generated by the oscillatory flow over a rippled bed depends on the parameters of the problem. In particular, these first simultations considered moderate values of the Reynolds number such that the flow regime can be assumed to be laminar.
 
 
 
[[Image: WaveRippleFormationFig14.jpg|thumb|500px|right|Figure 15. Time evolution of the crests of the ripples generated by the growth of a sandy Gaussian hump which interacts with an oscillatory flow characterized by <math>U_0=0.26</math> m/s, <math>T=6</math> s, <math>d=0.3</math> mm (adapted from Marieu et al. <ref name=M></ref>.]]
 
 
 
Nowadays, the power of computers is such as to allow the evaluation of the turbulent flow field
 
by means of Direct Numerical Simulations (DNS) of Navier-Stokes and continuity equations or using Large Eddy Simulations (LES) (Scandura et al. <ref>Scandura P., Vittori G. and Blondeaux P. (2000). Three-dimensional oscillatory flow over steep ripples. J. Fluid Mech. 412, 355-378.</ref>, Barr and Slinn<ref>Barr B. and Slinn, D. (2004). Numerical simulation of turbulent, oscillatory flow over
 
sand ripples. J. Geophys. Res., 109 (C9), 1-19.</ref>, Zedler and Street<ref>Zedler E.A. and Street R.L. (2006). Sediment transport over ripples in oscillatory flow. J. Hydraul. Eng. A.S.C.E. 132 (2), 1-14</ref>, Chalmoukis and Dimas<ref>Chalmoukis I.A. and Dimas A.A. (2016). Numerical simulation of oscillatory flow over 3-D vortex ripples using the Immersed Boundary Method. 26th Int. Ocean and Polar Eng. Conf., 26 June-2 July, Rhodes, Greece. ISOPE-I-16-549.</ref>, Leftheriotis and Dimas<ref>Leftheriotis G. and Dimas A. (2016). Large Eddy Simulation of oscillatory flow and mor-
 
phodynamics over ripples. Proc. 35th Conference on Coastal Engineering, Antalya, Turkey,
 
2016.</ref>). However, for practical applications, it is convenient to compute the turbulent oscillatory flow over vortex ripples by using the Reynolds averaged equations and a turbulence model.
 
 
 
Andersen<ref>Andersen K.H. (1999). The dynamics of ripples beneath surface waves and topics in shell models of turbulence. PhD thesis, Det Naturvidens-kabelige Fakultet Københavns Universitet.</ref> was one of the first to couple the evaluation of the turbulent flow by means of a RANS approach with the evaluation of the time development of the bottom profile computed by means of Exner equation. More recently, Marieu et al. <ref name=M>Marieu V., Bonneton P., Foster D.L. and Ardhuin F. (2008). Modeling of vortex ripple morphodynamics. J. Geophys. Res. 113 (C09007), 1-15.</ref> used a similar approach to simulate ripple growth from a quasi-flat bed. Turbulence closure was achieved by means of the model of Wilcox<ref>Wilcox D.C. (1988). Re-assessment of the scale-determining equation for advanced turbulence models. AIAA Journal 26 (11), 1299-1310.</ref> while the sediment transport rate took into account both the bed load contribution and the suspended load contribution. Moreover, the morphology module simulated the avalanches that take place at the crests of the ripples when the steepness of the profile becomes too large. Figure 15 shows the time development of the position of the crests of the ripples generated by the interaction of an oscillatory flow with a Gaussian hump located at the middle of the computational domain, It clearly appears that further ripples are generated around the initial hump. Later these ripples are characterized by a complex nonlinear dynamics; they migrate, split, merge, annihiliate and they eventually attain an equilibrium configuration.
 
 
 
 
 
  
  
 
==Related articles==
 
==Related articles==
 
+
:[[Integrated Coastal Zone Management (ICZM)]]
[[Wave ripples]]
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:[[Thresholds of environmental sustainablility]]
 
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:[[Sustainability indicators]]
[[Stability models]]
 
 
 
[[Bedforms and roughness]]
 
 
 
[[Sand transport]]
 
 
 
[[Sediment transport formulas for the coastal environment]]
 
 
 
  
  
 
==References==
 
==References==
 
 
<references/>
 
<references/>
  
  
 +
<br>
  
{{2Authors
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{{author
|AuthorID1=14087
+
|AuthorID=11928
|AuthorFullName1= Paolo Blondeaux
+
|AuthorFullName=Vos, Matthijs
 
+
|AuthorName=Matthijs}}
|AuthorID2=14090
 
|AuthorFullName2= Giovanna Vittori
 
  
}}
+
[[Category:Coastal and marine ecosystems]]
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[[Category:Integrated coastal zone management]]
[[Category:Land and ocean interactions]]
 
[[Category:Geomorphological processes and natural coastal features]]
 
[[Category:Coastal processes, interactions and resources]]
 
[[Category:Coastal and marine natural environment]]
 

Revision as of 14:04, 24 August 2020



Resilience and resistance


Definition of Resistance:
The capacity to weather a disturbance without loss (Lake 2013[1]).
This is the common definition for Resistance, other definitions can be discussed in the article


Definition of Resilience:
(1) the capability to anticipate, prepare for, respond to, and recover from significant multihazard threats with minimum damage to social well-being, the economy, and the environment (sometimes called 'socio-ecological resilience')(Olsen et al. 2019[2]);

(2) the capability of a (socio-)ecological system to remain within a stability domain when subjected to environmental change, while continually changing and adapting yet remaining within critical thresholds (sometimes called 'general resilience') (Folke et al. 2010[3]; Scheffer 2009[4]; Brand and Jax 2007[5]);

(3) the capacity to experience shocks while retaining essentially the same function, structure, feedbacks, and therefore identity (sometimes called 'ecological resilience') (Brand and Jax 2007[5]; DEFRA 2019[6]), which is closely related to the concept of 'ecosystem resistance': the amount of disturbance that a system can withstand before it shifts into a new regime or an alternative stable state (Holling 1973[7]; Gunderson 2000[8]);

(4) the capacity of an ecosystem to regain its fundamental structure, processes, and functioning (or remain largely unchanged) despite stresses, disturbances, or invasive species (e.g., Hirota et al., 2011[9]; Chambers et al., 2014[10]; Pope et al., 2014[11]; Seidl et al., 2016[12]), which can be measured by the time needed to recover its original state (sometimes called 'engineering resilience'[1]).
This is the common definition for Resilience, other definitions can be discussed in the article


Introduction

Coastal and marine ecosystems are affected by environmental disturbance at a variety of spatio-temporal scales. The organisms inhabiting these systems are adapted to such disturbance, either by being tolerant of these conditions or by playing a role in one or more of the successional stages that follow during ecosystem recovery.

If all species in the system were tolerant to a particular perturbation, very little would change at the ecosystem level, and we could call the system resistant to this disturbance. However, often a disturbance, such as a temporary very low oxygen level, affects a substantial proportion of the organisms dramatically, either causing them to die, or forcing them to rapidly migrate to more favorable parts of the environment. Such an adverse disturbance could locally defaunate a certain volume in the pelagic or a certain area of hard or soft substrate. Such destruction at a local scale does not mean the end of local functioning. Usually organisms are available at a larger spatial scale that can re-colonize the affected area, according to their particular tolerances and abilities to favorably affect their local environment.

The term resilience has been defined in different ways, illustrated in the definition above. According to DEFRA (2019[6]) there is limited consensus in the literature about how resilience can be characterized and assessed. The term resilience is sometimes used to represent some kind of normative proposition about what kinds of ecosystem characteristics are desirable or necessary in the context of sustainable development, reflecting particular cultural and philosophical assumptions[6]. However, the resistance of an ecosystem (see the definition above) to changing conditions and the rate of recovery following some disruptive event are generally considered major components of resilience that can in principle be expressed in quantitative terms.

Other attributes such as the capacity of ecosystems to transform and adapt in the face of environmental change (i.e. system's ability to re-organize itself) are more difficult to translate to practice. According to Dawson et al. (2010[13]), resilience concerns the response of ecosystems to changing environmental conditions and must be looked at alongside other additional dynamic features, namely durability, robustness and stability. These concepts can be defined as[13]:

  • Durability: ability to cope with a chronic stress, but the source of this stress is endogenous;
  • Robustness: ability to recover or maintain the systems' social-ecological functions in the face of an external and chronic driver;
  • Stability: system’s tolerance to transient and endogenous shocks or disruptions.

Both resistance and resilience cause an ecosystem to remain relatively unchanged when confronted to a disturbance, but in the case of resistance no internal re-organization and successional change is involved. In contrast, resilience implies that the system is internally re-organizing, perhaps through a mozaic of patches that are at different stages of re-assembly. System responses to changing environmental conditions are displayed schematically in Fig. 1, corresponding to different resilience characteristics.

Figure 1. Schematic representation of the trajectories of a (socio-)ecological system in a plane defined by the system state (fundamental structure, processes, and functioning - vertical axis) and the change of environmental conditions (horizontal axis), for different resilience characteristics (a, b, c, d). The initial state corresponds to the position on the graph at the vertical axis (zero change in environmental conditions). In all situations the ecosystem is assumed to collapse irreversibly (down to the horizontal axis) when the change in environmental conditions is much greater than the systems' resistance. The angle [math]\alpha[/math] represents the rate at which the system recovers when the change in environmental conditions is reduced (small [math]\alpha[/math] means slow recovery, large [math]\alpha[/math] means fast recovery). Panel a: Resilience characterized by high resistance (definition 3) and slow recovery (definition 4). Panel b: Resilience characterized by low resistance and fast recovery. Panel c: Resilience characterized by a shift to an alternative stable system state. Panel d: Low resilience, characterized by low resistance and slow recovery.


When considering the potential effect of a certain type of disturbance it is thus useful to ask two questions:

  1. Will the species of this system be able to tolerate it (implying resistance), and if not,
  2. Is recovery possible through a successional trajectory, back to the same, or at least a desirable, ecosystem state (implying resilience)?

Resistance breaks down when uni-directional ongoing change acts faster than the organisms' ability to adapt their tolerances. If uni-directional ongoing change is this fast (even if gradual), the system will not be sufficiently resilient either, as full recovery through succession will then not be possible. Recovery from sudden and local disturbance is often possible through recolonization, but the rate of recovery will depend crucially on the spatial extent of disturbance. For example, recovery from anoxia could take 5 to 8 months at the scale of square meters (Rossi et al. 2009[14]), but could take 5 to 8 years at the scale of a whole bay (Diaz & Rosenberg 1995[15]).

According to definition (4), the speed at which an ecosystem returns to its former state following a (minor) disturbance can be considered a measure of resilience. The idea is that a system with a short return time is more resilient than one with a long return time. Such resilience measured as (1 / the return time to a stable equilibrium) has also been called engineering resilience. It has however a long history of use among ecologists (Pimm 1982[16], DeAngelis 1992[17], Vos et al. 2005[18]). Resilience is also used in a way that more closely resembles the definition of resistance. Ecological resilience was defined as the amount of disturbance that an ecosystem could withstand without changing self-organized processes and structures (definition 3).

Resilience of coastal systems largely depends on biodiversity, which is a major requirement for allowing ecosystems to adapt to changing conditions. The human impact on the environment through pollution, fisheries, sediment erosion / deposition and global climate change has brought about much faster change than would occur under natural conditions, putting severe stress on many ecosystems. Without genetic diversity, natural selection cannot occur and if natural selection is limited, adaptation is impossible. Preservation of biodiversity and, more specifically, genetic diversity is therefore of paramount importance for successful adaptation to our rapidly changing environments. However, biodiversity may not always protect ecosystems from major abiotic disturbances (Folke et al. 2004[19]).

Resilience through recolonization

To understand resilience of ecosystems it is essential to understand what drives succession within these ecosystems. Succession determines how, and how fast, communities return to their original state, or perhaps enter a new state. Many aspects of succession can be understood in terms of trade-offs between the ability to be either a good early (re)colonizer, or a good competitor. Succession involves a gradual replacement of colonizer/competitor species according to the degree to which they tolerate, facilitate or inhibit certain environmental conditions and other species (Rossi et al. 2009[14]). The extent to which processes of (re)colonization and succession can take place largely determines the recovery of ecosystems after major disruption and is therefore an essential characteristic of the resilience of ecosystems.

In this context, it is important to consider the spatial component of ecosystem resilience. Diversity of structurally and functionally connected landscapes, rich in resources and species, promotes the flow or movement of individuals, genes, and ecological processes. Below certain thresholds of connectivity the capacity to regain structure and function after perturbation is lost (Holl and Aide, 2011; Rudnick et al., 2012;McIntyre et al., 2014; Rappaport et al., 2015; Ricca et al., 2018). Chambers et al. (2019[20]), based on Allen et al. (2016[21]), have therefore introduced the concept of 'spatial resilience', which is a measure of how spatial attributes, processes, and feedbacks vary over space and time in response to disturbances and affect the resilience of ecosystems. Self-organization through strong feedbacks at multiple scales and high levels of functional diversity and redundancy, stabilizes the system with respect to disturbances within the range of historic variability.

When creating Marine Protected Areas, the sources of populations at all stages of succession should be protected, to preserve 'ecological memory' to the fullest possible extent. This includes protecting not only 'high quality' habitats that harbour healthy mature communities, but also 'low quality' and disturbed habitats that are required for those species that contribute to early recovery of perturbed areas (Rossi et al. 2009[14]). The selection of Marine Protected Areas thus involves evaluating the number, size, and spatial configuration of habitat fragments and degree of connectivity required to support restoration of ecosystems and conservation of focal habitats and species[20][2].

Resistance to changes in abiotic and biotic factors

Community composition and ecosystem function may change very little under environmental change when the organisms can adapt to such change or tolerate it for some time (when the change is only temporary). However, all organisms have bounds to what they can temporarily or permanently tolerate, and when change exceeds some of these limits, the community composition and ecosystem functioning is likely to change.

It is unlikely that communities can be resistant to ongoing gradual change, such as global warming. Acclimation and phenotypic plasticity do not suffice to maintain the system as it is. Genetic adaptation could allow community members to track such abiotic environmental change, but it is more likely that the area where the community is functioning will be invaded by species that function well at higher temperatures. The original species will thus have to deal with new competitors and predators, in addition to the changed abiotic factor. To some extent the original community can track the preferred temperature range, by moving spatially to greater depths or to alternative geographic areas. But these new areas are likely to differ in other ecological aspects such as water pressure, light climate and perhaps speeds of water flow etc.

Adaptation and the consequences of mortality at different trophic levels

External disturbance interacts with internal mechanisms that shape community structure. To understand how an increased mortality of top-predators will affect the entire food chain, it is essential to understand how processes of mutual adaptation within food chains already give shape to existing patterns such as trophic structure (how biomass in ecosystems is partitioned between trophic levels).

Abundances at different trophic levels (such as algae, herbivores, carnivores and top-predators) and their responses to increased mortality (as under environmental change) depend critically on different mechanisms of adaptation within food chains and on the importance of population density at each of these trophic levels. However, different types of adaptation to living in a food chain context (balancing the need to acquire resources with the need to avoid predation) can often have similar consequences. For example, micro-evolution of behaviour, species replacement and induced defenses at a middle trophic level may all have similar effects on trophic level abundances in disturbed food chains (Abrams and Vos 2003[22]).


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The main author of this article is Vos, Matthijs
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Citation: Vos, Matthijs (2020): Testpage3. Available from http://www.coastalwiki.org/wiki/Testpage3 [accessed on 29-03-2024]