**1. Introduction**

Equilibrium is a term having many different meanings. In physics, it describes the average condition of a system, as measured through one of its elements or attributes over a specific period of time. However, the geomorphological concept of equilibrium has many confusing meanings [1], e.g., see below plus, quasi-equilibrium, time independence, and its semantics have been seemingly lost in a vast array of papers, e.g., the authors of [2–5] showed that catastrophe theory even suggests that any system might have numerous equilibrium states.

Equilibrium theory arises from Newton's laws of motion (F = MLT−2) and refers to where an object's velocity is constant (no acceleration), or if an object is stationary (at rest) and any force acting on it has its vector sum as zero, i.e., force and reaction are balanced and the system's properties are unchanged over time. This is static equilibrium. Dean ([6], p.399), referring to beach dynamic equilibrium, defined it as "the tendency for beach geometry to fluctuate about an equilibrium, which also changed through time, but much more slowly." He added that our ability to predict quantitatively these changes is likely to remain poor for the coming decades.

A system can also exhibit other states:


**Citation:** Pranzini, E.; Williams, A.T The Equilibrium Concept, or . . . (Mis)concept in Beaches. *Geosciences* **2021**, *11*, 59. https://doi.org/ 10.3390/geosciences11020059

Academic Editors: Jesus Martinez-Frias and Gianluigi Di Paola Received: 30 December 2020 Accepted: 25 January 2021 Published: 29 January 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Further readings about these states may be found in [7,8].

Additionally, Renwick [9] has argued that if one has equilibrium (absence of a discernible trend over the period under observation), then its inverse is a disequilibrium state, i.e., a landform that tends towards equilibrium, but has not had sufficient time to reach it, i.e., a decaying state. The end result is confusion. It is not only with respect to beaches that the term is frequently used; care should be taken in usage of the term, as it has been strewn around the literature in many guises, e.g., equilibrium shoreline evolution models [10]; equilibrium types in planforms of bay beaches [11]. The latter's findings ranged from "dynamic" planforms where there is a constant sediment throughput to maintain beach stability, to a "static" position, as a result of a reduced/ceasing of an updrift sediment supply. Jackson and Cooper ([11], p.112), further commented that "Static equilibrium models represented a convenient yardstick with which to ascertain a particular shoreline's current stability status." However, they urged caution to the approach in identifying equilibrium and non-equilibrium shorelines, mentioning reliance on contemporary beach morphometrics as an input, and omission of other dynamic variables (secondary wave motions, tidal, and river currents).

Its geomorphological origin can be traced to Gilbert's (1877) classic work on sediment flux at the drainage basin scale in the Henry Mountains, USA [12], later quantified by Ahnert [13], relating to rock erosion and resistance, the concept of a systems approach in geomorphology and negative feedback. It is a process-based approach, and most discussion of the term has been in reference to hydrological processes and concepts. Gilbert [12] argued that all streams worked towards a graded condition (dynamic equilibrium) where the net effect of the stream is neither erosion nor deposition. Willgoose et al. [14] expanded this arguing that within river catchments episodic fluctuations can occur, but over time in dynamic equilibrium, a balance exists between uplift and erosion.

Equilibrium in any system has to be time dependent but the particular attributes must be defined over a particular period, but how is the scale defined? A classic paper on the fundamental importance of scale was that of Schumm and Lichty [15]. They showed that input/output relationships could change within any timespan, leading to the concept of fast and slow variables. Beattie and Oppenheim [16], dealing with thermodynamics, argued that processes can produce different equilibrium states that take place so slowly that no discernible displacement in the presumed equilibrium state can detected. Is this true in geomorphology and particularly in a beach environment? Usage of the equilibrium term as being scale-dependent is perhaps questionable, as it depends on the observer's view of the term. Howard ([17], p.71) was of the opinion that, "Equilibrium systems are generally not applicable to physical systems that exhibit oscillatory, threshold, hysteretic, multivalued or strongly unpredictable responses to constant or slowly changing inputs." He further stated that a change of time/scale could make a system giving complex responses be predictable or in equilibrium. Bracken and Wainwright [18] even posed the question whether equilibrium was a myth or a metaphor, arguing for the latter? Is the concept testable? Richards [19] even suggested that it was impossible to measure equilibrium.

For beaches, which are the concern of this paper, equilibrium infers a lack of change in a system (a component set gathered into a whole), as inputs and outputs remain in balance. If changes do occur moving it to a new position (dynamic equilibrium), then feedbacks (an output causing system changes to the inputs) will allow for correction (Figure 1). Feedbacks can be positive, i.e., exacerbating the size of the systems normal elements through time, or negative that can dampen or reverse the size of the systems elements/attributes. It is assumed that self-regulation occurs. Any beach can be thought of as an open system, i.e., energy/matter relating to the sediment budget can arrive and depart and if these are in balance then equilibrium is assumed to exist [20], e.g., beach sediment arrival equating to sediment removal. When this is broken, e.g., not only by human intervention (damming rivers, coastal structures, etc.), but also by the natural system itself, as occurred before the appearance of man on Earth, it is argued that the system will change in order to bring it back to equilibrium.

If one analyses the above closely, three key points emerge:


Does this infer that beaches can never be stable?

Among the forms that make up the Earth's surface, beaches are certainly those subject to the most rapid variations. A storm, even a modest one, is enough for the beach to assume a different morphology. The shape and position of the shoreline varies, as does the slope of the swash zone, the seabed, the position, size, and number of submerged bars. Moreover, from a granulometric viewpoint, the changes are significant and frequent, e.g., a small hole dug on the shoreline allows one to see more or less coarse sand levels that have been deposited in the previous days in different marine weather conditions.

Taking into consideration the whole beach (emerged and submerged part), these variations, both morphological and granulometric, are mainly due to movement of sand to and from the shore, and the overall volume of sediments that make up the beach does not vary as consistently as the changes in its morphology might suggest. For a short time scale, the beach is subject to feedback processes that make it assume the form that best dissipates wave energy, and as this changes moment by moment, even the beach adapts and changes. However, in these short periods some sediment can enter/exit, but does this infer that a beach is stable? Ahnert's ([21], p.322) comment that, "the huge concentration of energy in littoral processes means that all artificial disturbances of the natural dynamic equilibrium (sic) on the seacoast have very rapid and often unforeseen effects," is particularly pertinent here.

Beach stability is a concept frequently present in research papers, official documents and the press. Generally, claiming that beaches are not stable relates to concern for coastal sectors experiencing erosion, as accretion, in most cases, is not a problem. For the press and general public out of the three possible conditions for a beach, i.e., accretion, erosion, and stability, the latter is frequently considered as the natural one and the two-remaining are attributed to disturbances governed by anthropogenic actions, although beaches appeared, disappeared, grew and reduced before the appearance of *Homo sapiens.*

Hourly, daily, and seasonally changes in dry beach width are recognized in every coast, due to tide, air pressure, and waves, but these are considered as "normal" oscillations around longer-term "stable" conditions. Their magnitude order can range from few centimeters in micro-tidal sheltered coasts, e.g., many Mediterranean beaches, to hundreds of meters in macro-tidal exposed environments, e.g., the Glamorgan Heritage Coast, Wales, UK. Many papers/books refer to the sweep zone or envelope: a large sweep zone delineating an unstable beach, a very narrow one a stable (equilibrium) beach.

Bogaert et al. [22] distinguished between behavior and evolution. However, it is not the shoreline position which determinates beach condition but the beach sediment budget, calculated for a selected coastal segment and extending from the first dune (or anthropogenic structure) to the depth of closure [23]. The problem of high-frequency beach morphological changes around a hypothetical equilibrium has frequently been resolved by referring to the concept of dynamic equilibrium [6], but this status cannot be attributed to a beach as it refers to a continuous process which does not change the shape, temperature, or chemical constitution of an object. In chemistry, dynamic equilibrium is when, e.g., substances transition between the reactants and products at equal rates, meaning there is no net change, and can only occur in reversible reactions, i.e., when the rate of the forward reaction is equal to the rate of the reverse reaction.

The equilibrium concept could be applied to pocket beach rotation [24], where grains eroded on one side are deposited on the other one, but also in this case the balance between input and output is questionable. What is under discussion here is the existence of that hypothetical equilibrium around which the beach should fluctuate. The approach to this problem in this paper starts with processes that form the beach sediment budget looks at how they can modify sediment input and output from the beach, considers whether related processes are feedback regulated or not, and, finally, to try to find a functional definition of beach stability (if any).

A stable beach is one that oscillates "slightly" around a mean, as against a high oscillatory component that can spill into an erosion/deposition mode, i.e., an unstable beach has the potential to enter this phase. The time factor must be long enough to dampen out any irregularities due to transient storm activity. Many studies rely on the estimation done via interviews by Bird [25] who found that over 70% of shorelines are retreating because of climate change related processes, and this trend is increasing [26]. If these shorelines are in an eroding mode, then they are by definition unstable [26]. It should be noted that the figure quoted should refer to beaches that have been studied, as all beaches in the world were not investigated. Alternatively, Luijendijk et al. [27], through machine learning and image processing techniques on 1.9 million historical Landsat images, carried out global scale studies concluding that 24% of the beaches are eroding at a rate >0.5 m/year. Applying the same method, 48% of world beaches resulted as stable, but within a range of ±0.5 m/year.
