**3. Conclusions**

Most of the above examples dealt with in this paper are not linked by feedback processes, e.g., an eroding beach cannot send a message upstream to ask the river for more sediment. Without feedback any equilibrium is impossible not only in nature, but also in, for example, economics or human sciences. The beach is subject to feedbackregulated processes [121], that act within the beach and do not operate linking it with the external environment (atmosphere, catchment, etc.). Beach slope changes under the attack of different waves (to better dissipate wave energy), bars formed at the breaker line induce more waves to break there (a positive feedback) but waves can demolish the bar if it is too elevated (negative feedback) giving it a well-defined height.

Even without considering variations in each of these input and output items, each of which involves from few to millions of cubic meters of sediments a year, it is statistically impossible that their algebraic sum is zero, or even not far from zero. When a beach is classified to be in equilibrium, it almost certainly means that data could not reveal changes that were of the same order of magnitude; and one may question the data accuracy.

For the Tuscany coast a stable classification refers to sectors where the mean shoreline displacement of two surveyed shorelines was between +/−5 m. As far as the real sedimentary budget is considered, i.e., the sand volume change, the surveys accuracy is generally unsuitable to monitor actual values. Echo sounding accuracy is between 5 and 10 cm, and in the worst conditions depth changes of +/−20 cm are not significant [122]. If the depth of closure is 1 km offshore, 200 m<sup>3</sup> per meter of coast is the expected error, which equals a medium artificial nourishment project.

Assessing the possibility for a beach to be in equilibrium is contentious and socially dangerous. To give stakeholders this idea makes them ask for this stability and turns away the possibility of allowing the beach to achieve its due resilience. In many papers, "stability" has been used, but clearly exhibiting such a beach evolution trend refers to cases where observed changes are within any measurement error.

Many coastal geomorphic features are relatively stable and form changes take place slowly; others are less stable and have comparatively rapid change in response to dynamic environmental factors [123]. Quickly changing trends are a major problem for the present, and for the temporal dimension, which we usually use to build the knowledge, explanations, and predictions [44,124].

Unconsolidated beach materials respond rapidly to changes in the dynamic environment. Comments made in many books and documents of the last century, were to the effect that if a beach for an adequate period has been subject to environmental forces, then the profile will respond to both long-term and short-term changes, which tends to restore an equilibrium profile. The view was that equilibrium—the amount of sediment deposited by waves and currents—will be balanced by the amount removed, with many researchers even producing equations for the slope of natural profiles [125]. Cooper and Pilkey ([126], p. 605) concluded an overview of long-shore transport modeling, assessing, "that our present understanding permits only a qualitative estimate of direction of long-shore drift and identification of some of the controls"; but are the controls reliable?

So what is beach stability and how is it (if at all) entwined with dynamic equilibrium, or has the term dynamic been corrupted and used as a useful blanket adjective? As has been shown, it is virtually impossible for subsequent beach profiling to cover the exact same profile, so the stability (equilibrium) concept must be questioned. A beach is certainly a dynamic entity as it constantly changes shape as sediment is moved around. Is dynamic equilibrium simply an idea for a beach striving to assume an ideal shape? Is stability neither erosion nor deposition? Is an unstable beach one undergoing strong erosion only—if so what about strong deposition? Subjectivity of input selection, survey accuracy and also the potential for geological, dynamic, and sedimentological constraints reflect the uncertainty in the quest for the elusive conundrum of equilibrium. Is it akin to Lewis Carroll's [127] snark, is it a Boojum (a particularly nasty type of snark, which if glimpsed by a person, he/she disappears, so perhaps the term might disappear from the literature), or an allegory of an attempt to discover an absolute measure that is doomed to failure? The concept is so theoretical as to be functionally irrelevant.

To end, we believe that the word "stability" should not be used for beaches, unless it is accompanied by a value of the accuracy with which this condition is assessed and the time period to which it refers.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
