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Nevanlinna theory

In the mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called it "one of the few great mathematical events of (the twentieth) century." The theory describes the asymptotic distribution of solutions of the equation f(z) = a, as a varies. A fundamental tool is the Nevanlinna characteristic T(r, f) which measures the rate of growth of a meromorphic function.

Other main contributors in the first half of the 20th century were Lars Ahlfors, André Bloch, Henri Cartan, Edward Collingwood, Otto Frostman, Frithiof Nevanlinna, Henrik Selberg, Tatsujiro Shimizu, Oswald Teichmüller, and Georges Valiron. In its original form, Nevanlinna theory deals with meromorphic functions of one complex variable defined in a disc |z| ≤ R or in the whole complex plane (R = ∞). Subsequent generalizations extended Nevanlinna theory to algebroid functions, holomorphic curves, holomorphic maps between complex manifolds of arbitrary dimension, quasiregular maps and minimal surfaces.

This article describes mainly the classical version for meromorphic functions of one variable, with emphasis on functions meromorphic in the complex plane. General references for this theory are Goldberg & Ostrovskii, Hayman and.

Nevanlinna's original definition
Let f be a meromorphic function. For every r ≥ 0, let n(r,f) be the number of poles, counting multiplicity, of the meromorphic function f in the disc |z| ≤ r. Then define the Nevanlinna counting function by

 * $$ N(r,f) = \int\limits_0^r\left( n(t,f) - n(0,f) \right)\dfrac{dt}{t} + n(0,f)\log r.\,$$

This quantity measures the growth of the number of poles in the discs |z| ≤ r, as r increases. Explicitly, let a1, a2, ..., an be the poles of ƒ in the punctured disc 0 < |z| ≤ r repeated according to multiplicity. Then n = n(r,f) - n(0,f), and

 * $$ N(r,f) = \sum_{k=1}^{n} \log \left( \frac{r}{|a_k|}\right) + n(0,f)\log r .\,$$

Let log+x = max(log x, 0). Then the proximity function is defined by

 * $$ m(r,f)=\frac{1}{2\pi}\int_{0}^{2\pi}\log^+ \left| f(re^{i\theta})\right| d\theta. \,$$

Finally, define the Nevanlinna characteristic by (cf. Jensen's formula for meromorphic functions)

 * $$T(r,f) = m(r,f) + N(r,f).\,$$

Ahlfors–Shimizu version
A second method of defining the Nevanlinna characteristic is based on the formula

 * $$ \int_0^r\frac{dt}{t}\left(\frac{1}{\pi}\int_{|z|\leq t}\frac{|f'|^2}{(1+|f|^2)^2}dm\right)=T(r,f)+O(1), \,$$

where dm is the area element in the plane. The expression in the left hand side is called the Ahlfors–Shimizu characteristic. The bounded term O(1) is not important in most questions.

The geometric meaning of the Ahlfors—Shimizu characteristic is the following. The inner integral dm is the spherical area of the image of the disc |z| ≤ t, counting multiplicity (that is, the parts of the Riemann sphere covered k times are counted k times). This area is divided by $\pi$ which is the area of the whole Riemann sphere. The result can be interpreted as the average number of sheets in the covering of the Riemann sphere by the disc |z| ≤ t. Then this average covering number is integrated with respect to t with weight 1/t.

Properties
The role of the characteristic function in the theory of meromorphic functions in the plane is similar to that of

 * $$\log M(r, f) = \log \max_{|z|\leq r} |f(z)| \,$$

in the theory of entire functions. In fact, it is possible to directly compare T(r,f) and M(r,f) for an entire function:

 * $$T(r,f) \leq \log^+ M(r,f) \,$$

and

 * $$\log M(r,f) \leq \left(\dfrac{R+r}{R-r}\right)T(R,f),\,$$

for any R > r.

If f is a rational function of degree d, then T(r,f) ~ d log r; in fact, T(r,f) = O(log r) if and only if f is a rational function.

The order of a meromorphic function is defined by

 * $$\rho(f) = \limsup_{r \rightarrow \infty} \dfrac{\log^+ T(r,f)}{\log r}.$$

Functions of finite order constitute an important subclass which was much studied.

When the radius R of the disc |z| ≤ R, in which the meromorphic function is defined, is finite, the Nevanlinna characteristic may be bounded. Functions in a disc with bounded characteristic, also known as functions of bounded type, are exactly those functions that are ratios of bounded analytic functions. Functions of bounded type may also be so defined for another domain such as the upper half-plane.

First fundamental theorem
Let a ∈ C, and define

\quad N(r,a,f) = N\left(r,\dfrac{1}{f-a}\right), \quad m(r,a,f) = m\left(r,\dfrac{1}{f-a}\right).\,$$

For a = ∞, we set N(r,∞,f) = N(r,f), m(r,∞,f) = m(r,f).

The First Fundamental Theorem of Nevanlinna theory states that for every a in the Riemann sphere,

 * $$ T(r,f) = N(r,a,f)+m(r,a,f) + O(1),\,$$

where the bounded term O(1) may depend on f and a. For non-constant meromorphic functions in the plane, T(r, f) tends to infinity as r tends to infinity, so the First Fundamental Theorem says that the sum N(r,a,f) + m(r,a,f), tends to infinity at the rate which is independent of a. The first Fundamental theorem is a simple consequence of Jensen's formula.

The characteristic function has the following properties of the degree:

 * $$\begin{array}{lcl}

T(r,fg)&\leq&T(r,f)+T(r,g)+O(1),\\ T(r,f+g)&\leq& T(r,f)+T(r,g)+O(1),\\ T(r,1/f)&=&T(r,f)+O(1),\\ T(r,f^m)&=&mT(r,f)+O(1), \, \end{array}$$

where m is a natural number. The bounded term O(1) is negligible when T(r,f) tends to infinity. These algebraic properties are easily obtained from Nevanlinna's definition and Jensen's formula.

Second fundamental theorem
We define $\overline{N}$(r, f) in the same way as N(r,f) but without taking multiplicity into account (i.e. we only count the number of distinct poles). Then N1(r,f) is defined as the Nevanlinna counting function of critical points of f, that is

 * $$ N_1(r,f) = 2N(r,f) - N(r,f') + N\left(r,\dfrac{1}{f'}\right) = N(r,f) + \overline{N}(r,f) + N\left(r,\dfrac{1}{f'}\right).\,$$

The Second Fundamental theorem says that for every k distinct values aj on the Riemann sphere, we have

 * $$ \sum_{j=1}^k m(r,a_j,f) \leq 2 T(r,f) - N_1(r,f) + S(r,f). \,$$

This implies

 * $$ (k-2)T(r,f) \leq \sum_{j=1}^k \overline{N}(r,a_j,f) + S(r,f),\,$$

where S(r,f) is a "small error term".

For functions meromorphic in the plane, S(r,f) = o(T(r,f)), outside a set of finite length i.e. the error term is small in comparison with the characteristic for "most" values of r. Much better estimates of the error term are known, but Andre Bloch conjectured and Hayman proved that one cannot dispose of an exceptional set.

The Second Fundamental Theorem allows to give an upper bound for the characteristic function in terms of N(r,a). For example, if f is a transcendental entire function, using the Second Fundamental theorem with k = 3 and a3 = ∞, we obtain that f takes every value infinitely often, with at most two exceptions, proving Picard's Theorem.

Nevanlinna's original proof of the Second Fundamental Theorem was based on the so-called Lemma on the logarithmic derivative, which says that m(r,f'/f) = S(r,f). A similar proof also applies to many multi-dimensional generalizations. There are also differential-geometric proofs which relate it to the Gauss–Bonnet theorem. The Second Fundamental Theorem can also be derived from the metric-topological theory of Ahlfors, which can be considered as an extension of the Riemann–Hurwitz formula to the coverings of infinite degree.

The proofs of Nevanlinna and Ahlfors indicate that the constant 2 in the Second Fundamental Theorem is related to the Euler characteristic of the Riemann sphere. However, there is a very different explanations of this 2, based on a deep analogy with number theory discovered by Charles Osgood and Paul Vojta. According to this analogy, 2 is the exponent in the Thue–Siegel–Roth theorem. On this analogy with number theory we refer to the survey of and the book by.

Defect relation
The defect relation is one of the main corollaries from the Second Fundamental Theorem. The defect of a meromorphic function at the point a is defined by the formula

 * $$ \delta(a,f)=\liminf_{r \rightarrow \infty}\frac{m(r,a,f)}{T(r,f)} = 1 - \limsup_{r \rightarrow \infty} \dfrac{N(r,a,f)}{T(r,f)}. \, $$

By the First Fundamental Theorem, 0 ≤ δ(a,f) ≤ 1, if T(r,f) tends to infinity (which is always the case for non-constant functions meromorphic in the plane). The points a for which δ(a,f) > 0 are called deficient values. The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most countable and the following relation holds:

 * $$ \sum_{a}\delta(a,f)\leq 2, \,$$

where the summation is over all deficient values. This can be considered as a generalization of Picard's theorem. Many other Picard-type theorems can be derived from the Second Fundamental Theorem.

As another corollary from the Second Fundamental Theorem, one can obtain that

 * $$ T(r,f')\leq 2 T(r,f)+S(r,f),\,$$

which generalizes the fact that a rational function of degree d has 2d − 2 < 2d critical points.

Applications
Nevanlinna theory is useful in all questions where transcendental meromorphic functions arise, like analytic theory of differential and functional equations holomorphic dynamics, minimal surfaces, and complex hyperbolic geometry, which deals with generalizations of Picard's theorem to higher dimensions.

Further development
A substantial part of the research in functions of one complex variable in the 20th century was focused on Nevanlinna theory. One direction of this research was to find out whether the main conclusions of Nevanlinna theory are best possible. For example, the Inverse Problem of Nevanlinna theory consists in constructing meromorphic functions with pre-assigned deficiencies at given points. This was solved by David Drasin in 1976. Another direction was concentrated on the study of various subclasses of the class of all meromorphic functions in the plane. The most important subclass consists of functions of finite order. It turns out that for this class, deficiencies are subject to several restrictions, in addition to the defect relation (Norair Arakelyan, David Drasin, Albert Edrei, Alexandre Eremenko, Wolfgang Fuchs, Anatolii Goldberg, Walter Hayman, Joseph Miles, Daniel Shea, Oswald Teichmüller, Alan Weitsman and others).

Henri Cartan, Joachim and Hermann Weyl and Lars Ahlfors extended Nevanlinna theory to holomorphic curves. This extension is the main tool of Complex Hyperbolic Geometry. Henrik Selberg and Georges Valiron extended Nevanlinna theory to algebroid functions. Intensive research in the classical one-dimensional theory still continues.

title: Nevanlinna theory


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Whithorn

Whithorn (Taigh Mhàrtainn), is a royal burgh in the historic county of Wigtownshire in Dumfries and Galloway, Scotland, about 10 mi south of Wigtown. The town was the location of the first recorded Christian church in Scotland, Candida Casa "White/Shining House", built by Saint Ninian about 397 CE.

Toponymy
There is a tradition that St Ninian built a church of stone and lime nearby in the late 4th century; it was called Candida Casa, 'White/Shining House'. "Whithorn" is a modern form of the Anglo-Saxon version of this name, Hwit Ærn or Hwiterne, 'White House'. In Gallovidian Gaelic, it was called Rosnat, or Futarna, the latter a version of the Anglo-Saxon name (Gaelic has no sound corresponding to English wh). Ninian dedicated the church to his master Martin of Tours, and when he died (probably in 432) Ninian was buried in the church.

Early history
A monastery and diocese of the Anglo-Saxon kingdom of Northumbria was founded on the site in the 8th century, possibly originating with a 6th-century Magnum Monasterium, or monastery of Rosnat. It was the centre of the revived See of Galloway (or Candida Casa) under the patronage of Fergus, Lord of Galloway and Bishop Gille Aldan from the 12th century. The late-medieval cathedral Whithorn Priory is ruinous, much of it having disappeared completely apart from the much-altered aisleless nave and vaults at the former eastern end which once held the shrine of St. Ninian, one of medieval Scotland's major pilgrimage destinations. A museum in the town contains finds from the site, which has been extensively excavated in recent years. A late-medieval gateway with the arms of the King of Scots leads into the site of the priory, which contains the 19th-century parish church and a museum of carved stones (Historic Environment Scotland). The collection of early medieval stones is one of the largest in Scotland, and includes the country's earliest surviving Christian memorial, the 5th-century inscribed Latinus Stone. The museum layout and display was revised and greatly improved in 2005.

The crozier
One of the finest artefacts found at the site is the Whithorn Crozier. The gilded and enamelled crozier is an outstanding example of champlevé enamels which were being made in England in the second half of the 12th century, and this one dates to around 1175. It is now housed in the National Museums of Scotland, although it is loaned to the Whithorn Trust Visitor Centre every summer. It is thought that the crozier was buried with the body of Simon de Wedale, who was one of the Bishops of Whithorn.

Churches

 * St Martin and St Ninian Catholic Church, George St, 1959–60. Designed by Harry Stuart Goodhart Rendel and is his only known building in Scotland. The interior was reordered with the altar brought forward from the east wall following the reforms of the Second Vatican Council. At that time the baldacchino was also demolished, and the decorative ironwork at the baptistry and communion rail scrapped. The original Creetown granite altar was placed outside against the east elevation. A possible source of inspiration is the church of San Julián de los Prados Ovieda, Asturias in Spain. The East elevation has a carved Hew Lorimer crucifix mounted to the wall. The crucifix has not fared well after poor cleaning in 1997 led to significant loss of detail. The church including the quadrant walls is listed Category C(S).
 * St Ninian's Priory Church, Church of Scotland. Built 1822 using stones from the medieval Whithorn Priory. The tower was added in the mid-19th century. Porch added by Peter MacGregor Chalmers in 1914.

Geography
Whithorn's link to the sea was the port known as the Isle of Whithorn (a separate community from Whithorn itself and actually a peninsula). It was much used in the Middle Ages by pilgrims arriving by boat. The thirteenth-century Saint Ninian's Chapel marked the point where pilgrims came ashore (the roofless remains are looked after by Historic Environment Scotland).

The 1st-century settlement of Rispain Camp, about 1 mi west of Whithorn, is also in the care of Historic Scotland.

Railway
Whithorn was once served by a railway station until 1950 when the passenger service was withdrawn and the freight services falling victim to the Beeching axe in 1964. The track was lifted in April 1965.

Listed buildings
List of listed buildings in Whithorn, Dumfries and Galloway

Residents

 * Charles Lockhart (1818–1905), petroleum producer and refiner, was born in Cairnhead, Isle of Whithorn. Co-founder in 1874 with John D Rockefeller of the Standard Oil Company (Esso).
 * The poet and scholar of South American literature Alastair Reid was born in Whithorn on 22 March 1926. He was known for his lighthearted style of poems and for his translations of South American poets Jorge Luis Borges and Pablo Neruda. Although he was known for translations, his own poems had gained notice during his lifetime. In his later years he was a frequent summer visitor, with his wife Leslie staying in a cottage on the Galloway House estate.
 * George Dickie (1912–1951) was born and grew up in Whithorn. Using the nom de guerre of Jack Brent, he fought in the Spanish Civil War of 1936–38, against the fascist forces of General Franco. A memorial plaque was unveiled to him in 2006 on the former butcher's shop next to The Pend where he once worked as an assistant. His story is told by his nephew, John Dickie, in the 2012 biography Geordie's Story: A Life of Jack Brent.
 * Jeannie Donnan (1864–1942), "The Galloway Poetess", was born in Gatehouse of Fleet in Kirkcudbrightshire and later moved to Whithorn on George Street commemorated by a plaque. She wrote local poetry published as Hameland: The Poems of Jeannie Donnan, 1907; War Poems, 1915; The Hills of Hame, 1930, as well as in the Galloway Gazette.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Whithorn


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Isle of Whithorn

Isle of Whithorn (Port Rosnait in Gaelic) is one of the most southerly villages and seaports in Scotland, lying on the coast north east of Burrow Head, about three miles from Whithorn and about thirteen miles south of Wigtown in Dumfries and Galloway. Whithorn, (Taigh Mhàrtainn in Gaelic), is a former royal burgh in Wigtownshire, Dumfries and Galloway, with which Isle of Whithorn is frequently incorrectly amalgamated or confused. It is referred to locally as 'The Isle' - never ' the Isle of Whithorn'.

The village is the location of the long ruined 13th-century Saint Ninian's Chapel, previously a chapel linked to Whithorn Priory and a stopping off point for pilgrims landing on Isle Head and making their way to Whithorn. Although no longer a true island, John Ainslie's maps as late as 1782 and 1821 show the Isle as an island. The main street was originally a causeway, with the harbour located on what was then the true Isle.

The Isle has a long history of habitation: for example, a survey of the ground between the Isle village and Saint Ninian's Chapel revealed Clearance cairns and cultivation furrows, as well as the remains of a rectangular building occupying the crest of the low hill immediately to the west of the chapel, with possible steps leading downhill towards the east.

Community and amenities
A Community Council represents the village and a Church of Scotland church, previously a Free Kirk, sits on the old boatyard, a site extending into the harbour and threatened at exceptionally high tides. There are a bowling club, a basic football pitch and two play areas for the young. The village hall is well used: in August 2008, the community took over management control of this facility and it is now completely refurbished and home to Scotland's most southerly cinema, 'Machars Movies'. Following a substantial award from the Big Lottery Fund, a new tearoom, post office and shop selling gifts and foodstuffs, 'Saint Ninian's', was built alongside the hall, and this opened in October 2014.

The village has one pub, the Steam Packet Inn. A second pub, the Queen's Arms, was closed some years ago and now houses an art gallery and craft shop. Also on the harbour there is a fish shop and general store called Isle Sea Foods; upstairs are a launderette, a toilet and shower facilities. The old village school is now a private house, overlooking the Stinking Port, the bay on the other side of the Isle promontory. A granite war memorial is at the north end of the village just off Main Street, on a mound overlooking the bowling green, with twelve First World War names on it and five Second World War names. The regiments and ships are given for the First World War only.

Fishing is widely available on the lochs, local rivers Bladnoch and Cree, and on the sea itself by boat or from the shore. There are numerous trails and quiet roads for walking, horse riding or cycling. Signposted by the Whithorn Pilgrimage Trust, the Pilgrim's Way follows a route southwards from the Southern Upland Way through the Machars, passing through New Luce and the early Christian holy sites of Glenluce Abbey, the White Loch, Whithorn, Isle of Whithorn and St Ninian's Cave. The main part of the route from Glenluce to Whithorn is 25 miles (40 km) long.

History
Pliny the Elder died in AD 79 and in his Natural History he describes Britain, Ireland and the smaller islands off the coast, one being vectis and this has been identified with the Isle of Whithorn. The name Lucopibia as used by Ptolemy has also been linked with Whithorn.

This was a haunt for smugglers and a principal harbour for legitimate trade and transport; here you caught the Countess of Galloway, the steam packet which would take you to Liverpool and maybe a passage on an emigrant ship to the New World or Australia.

Isle Castle
The Isle Castle is one of the village's oldest buildings. The building is comparatively small and is nearly square in plan, although the principle of an L-shaped arrangement is retained. In the early 19th century it was the residence of the Superintendent of the Coastguard, Sir John Reid. The 'Captain's Garden' is an extension into the harbour, now with several houses on it, but originally a garden associated with the castle. Still called the Captain's Garden it is decorated with simulated man o' war gun ports.

Above the door of these quarters is a stone with the initials of Patrick Houston of Drummaston and his wife Margaret Gordon, together with the likely date of construction, 1674. This stone has been moved from the old entrance door of the main building. The angle turrets on the second floor are carried on three plain corbels and do not show at all in the inside of the building. A Gothic-style porch was once attached to the Drumelan burn side of the castle.

The Bysbie corn mill was located on the other side of the Drummulin Burn from the castle and a sizeable mill pond and sluice supplied the water power; the mill remains lie behind the Queen's Arms Hotel. A chalybeate 'Spa Well' was located below the castle as shown by the 1850s 6 inch to the mile OS map.

St Ninian's Chapel

The chapel was repaired and partly rebuilt in 1898 by the Marquess of Bute; its main features were preserved. The present ruins are rectangular, measuring internally 31' × 16' 6" (9.4 m × 5.0 m). This building was probably erected to replace a 12th to early 13th century chapel, whose foundations were found during excavations and consisted of a nave, 17' 6" × 16' 6" (5.3 m × 5.0 m) with a square chancel. The chapel stood within a contemporary enclosure wall, part of which is still standing. An outer enclosure may have once existed, best seen in oblique sunlight, defining a roughly oval area 100' × 110' (30 m × 34 m). This outer enclosure may be of the Celtic period. There are no records of the chapel's use as a burial ground.

St Ninian is chiefly remembered at Whithorn. However, many places in Scotland are associated with his name, such as Saint Ninian's Isle in the River Garnock near Kilwinning. He converted the Southern Picts and therefore journeyed throughout many areas in southern Scotland.

The Cairn
The Isle's most familiar landmark is the square, white tower known locally as 'The Cairn', that has been a navigational aid for hundreds of years.

From here there are views of the Kirkcudbright coast, the whole of the northern range of the English Lake District mountains, and to the south the rounded, mountainous form of the Isle of Man.

During the Second World War it was the site of a tracking station for anti-aircraft gunnery practice. The set-up was that an aircraft flew up and down in front of the guns, which would try to bring it down. Initially, radio-controlled 'Queen Bee' aircraft were used, and the flat concrete foundation by the white tower on the cairn, now the location of the Solway Harvester memorial stone, was laid down for one of the control huts. The memorial stone to the Solway Harvester is in the form of an anchor set in a granite block.

Isle Head
This promontory is nearly bisected by Chapel Port east and west. The artificial ditch of the Old Fort is still clearly visible on the approach to Isle Head.

There is a multivallate promontory fort on Isle Head, consisting of three ramparts and two ditches, with a probable fourth rampart now reduced to a slight scarp. On the seaward side of the promontory are intermittent stretches of rubble linking and using natural outcrops of rock; in the south-east corner are inner and outer faces, indicating the footings of a wall up to 3 yards wide. On the levelled summit area are slight indications of rubble walling and artificial scarping on the south and east sides; and in the north-east corner are the possible remains of a hut circle.

A small block of rig is clearly visible on Isle Head, lying between the path leading to the promontory and the ramparts and ditch of the Iron Age fort. The furrows are orientated from NE to SW across the promontory.

A witness cairn was erected in 1997 to commemorate the arrival of the Christian missionary St. Ninian to these shores, and pilgrims are encouraged to place stones with personal messages onto a rock cairn. These cairns are sited in the remains of a building used to house the Isle of Whithorn lifeboat.

The lifeboat building had a roof until well after the Second World War, although the station was closed in 1919, when a motor lifeboat was placed on station at Kirkcudbright. The Lifeboat Station was set up in 1869. During the 50 years that the lifeboat station was operating there were three lifeboats:
 * Charlie Peake (1869–1886), 7 launches and 10 lives saved
 * Henry and John Leighton (1886–1901), 12 launches and 22 lives saved.
 * George and Margaret (1901–1919), six launches and six lives saved.

The Solway Harvester Seat is a granite-hewn bench and is one of two local tributes to the seven-strong local crew of the fishing boat Solway Harvester which sank in a storm off the Isle of Man in January 2000. The other, an anchor set in a stone inscribed with the vessel's name, sits atop the nearby hill next to The Cairn.

The harbour
The harbour remains the main focal point of the village. The Isle's own fleet is engaged in light trawling and lobster fishing. A number of leisure craft are kept here and the Isle is increasingly being used as a stopping off location for pleasure traffic plying the west coast, and as a launching site for trailer based craft such as small angling boats and diving RIB's.

A pier was first erected in the 16th century, but a more substantial structure was erected in 1790 to advance commerce, such as the trade with Whitehaven and other English ports. In 1969 the pier was badly damaged by storm and was only rebuilt after a prolonged campaign by local people. The perch is a reef of rock that runs out from the west side of the mouth of the Isle bay; it can be a hazard to craft coming into the Isle and for many years the end was marked with an iron pole.

In the 19th century the Isle also had strong trade links with Ireland and the Isle of Man, and ships were being built as well as sailed from here. The Countess of Galloway, the steam packet, once regularly sailed from here to Liverpool. By the end of the 20th century trading vessels had given way almost entirely to yachts and other leisure craft.

The Wigtown Bay Sailing Club welcomes all visiting yachtsmen whether they arrive by sea or by land with trailered boats. The clubhouse is situated on the harbour and has all the usual facilities. The harbour dries three hours either side of high water, and there is a dedicated visitors' berth at the outer end of the quay. There are also two slipways for launching trailer boats or dinghies, one of which is at the inner end of the quay and is sheltered in all weathers. The other, which gives access into the bay, can be used at virtually any state of the tide. The club is active throughout the summer with dinghy or cruiser races organised for most weekends and some races mid-week.

An unusual feature of the harbour was a fish trap; a wall or dyke running across part of the upper portion of the inlet which trapped fish as the tide went out. This was still marked as late as the 1850s on the OS map.

The Wicker Man
Various scenes from the 1973 horror film The Wicker Man were filmed in and around Isle of Whithorn. The climax of the film was filmed at St Ninian's Cave and on the clifftops at Burrow Head. The co-owner of the Isle of Whithorn castle, Elizabeth McAdam Laughland, plus several other local people featured in various scenes.

title: Isle of Whithorn


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Quantum turbulence

Quantum turbulence is the name given to the turbulent flow – the chaotic motion of a fluid at high flow rates – of quantum fluids, such as superfluids. The idea that a form of turbulence might be possible in a superfluid via the quantized vortex lines was first suggested by Richard Feynman. The dynamics of quantum fluids are governed by quantum mechanics, rather than classical physics which govern classical (ordinary) fluids. Some examples of quantum fluids include superfluid helium (4He and Cooper pairs of 3He), Bose–Einstein condensates (BECs), polariton condensates, and nuclear pasta theorized to exist inside neutron stars. Quantum fluids exist at temperatures below the critical temperature $$T_c$$ at which Bose-Einstein condensation takes place.

General properties of superfluids
The turbulence of quantum fluids has been studied primarily in two quantum fluids: liquid Helium and atomic condensates. Experimental observations have been made in the two stable isotopes of Helium, the common 4He and the rare 3He. The latter isotope has two phases, named the A-phase and the B-phase. The A-phase is strongly anisotropic, and although it has very interesting hydrodynamic properties, turbulence experiments have been performed almost exclusively in the B-phase. Helium liquidizes at a temperature of approximately 4K. At this temperature, the fluid behaves like a classical fluid with extraordinarily small viscosity, referred to as helium I. After further cooling, Helium I undergoes Bose-Einstein condensation into a superfluid, referred to as helium II. The critical temperature $$T_c$$ for Bose-Einstein condensation of helium is 2.17K (at the saturated vapour pressure), while only approximately a few mK for 3He-B.

Although in atomic condensates there is not as much experimental evidence for turbulence as in Helium, experiments have been performed with rubidium, sodium, caesium, lithium and other elements. The critical temperature for these systems is of the order of micro-Kelvin.

There are two fundamental properties of quantum fluids that distinguish them from classical fluids: superfluidity and quantized circulation.

Superfluidity
Superfluidity arises as a consequence of the dispersion relation of elementary excitations, and fluids that exhibit this behaviour flow without viscosity. This is a vital property for quantum turbulence as viscosity in classical fluids causes dissipation of kinetic energy into heat, damping out motion of the fluid. Landau predicted that if a superfluid flows faster than a certain critical velocity $$v_c$$ (or alternatively an object moves faster than $$v_c$$ in a static fluid) thermal excitations (rotons) are emitted as it becomes energetically favourable to generate quasiparticles, resulting in the fluid no longer exhibiting superfluid properties. For helium II, this critical velocity is $$v_c \approx 60 \text{m/s}$$.

Quantized circulation
The property of quantized circulation arises as a consequence of the existence and uniqueness of a complex macroscopic wavefunction $$\Psi$$, which affects the vorticity (local rotation) in a very profound way, making it crucial for quantum turbulence.

The velocity and density of the fluid can be recovered from the wavefunction $$\Psi(\mathbf{x},t)$$ by writing it in polar form $$\Psi(\mathbf{x},t) = |\Psi(\mathbf{x},t)|e^{i\phi(\mathbf{x},t)}$$, where $$|\Psi|$$ is the magnitude of $$\Psi$$ and $$\phi$$ is the phase. The velocity of the fluid is then $$\mathbf{v}(\mathbf{x},t) = (\hbar/m)\nabla \phi$$, and the number density is $$n(\mathbf{x},t) = |\Psi|^2$$. The mass density is related to the number density by $$\rho(\mathbf{x},t) = mn$$, where $$m$$ is the mass of one boson.

The circulation $$\Gamma$$ is defined to be the line integral along a simple closed path $$C$$ within the fluid

$$\Gamma = \oint_C \mathbf{v} \cdot \mathbf{dr}$$

For a simply-connected surface $$S$$, Stokes theorem holds, and the circulation vanishes, as the velocity can be expressed as the gradient of the phase. For a multiply-connected surface, the phase difference between an arbitrary initial point on the curve $$C$$ and the final point (same as initial point as $$C$$ is closed) must be $$2\pi q$$, where $$q=0,1,2,\cdots$$ in order for the wavefunction to be single-valued. This leads to a quantized value for the circulation

$$\Gamma = \frac{\hbar}{m} \oint_C \nabla S \cdot \mathbf{dr} = q\kappa$$

where $$\kappa = h/m $$ is the quantum of circulation, and the integer $$q$$ is the charge (or winding number) of the vortex. Multiply charged vortices ($$q>1$$) in helium II are unstable and for this reason in most practical applications $$q=1$$. It is energetically favourable for the fluid to form $$q$$ singly-charged vortices rather than a single vortex of charge $$q$$, and so a multiply-charged vortex would split into singly-charged vortices. Under certain conditions, it is possible to generate certain vortices with a charge higher than 1.

Properties of vortex lines
Vortex lines are topological line defects of the phase. Their nucleation makes the quantum fluid's region to become a multiply-connected region. As given by Fig 2, density depletion can be observed near the axis, with $$\rho = 0 $$ on the vortex line. The size of the vortex core varies between different quantum fluids. The size of the vortex core is around $$a_0 = 10^{-10}\text{m} $$ for helium II, $$a_0 = 10^{-8}\text{m} $$ for 3He-B and for typical atomic condensates $$a_0 = 10^{-4}\text{m} $$. The simplest vortex system in a quantum fluid consists of a single straight vortex line; the velocity field of such configuration is purely azimuthal given by $$v_{\theta} = \kappa/{2\pi r}$$. This is the same formula as for a classical vortex line solution of the Euler equation, however, classically, this model is physically unrealistic as the velocity diverges as $$r \rightarrow 0$$. This leads to the idea of the Rankine vortex as shown in fig 2, which combines solid body rotation for small $$r$$ and vortex motion for large values of $$r$$, and is a more realistic model of ordinary classical vortices.

Many similarities can be drawn with vortices in classical fluids, for example the fact that vortex lines obey the classical Kelvin circulation theorem: the circulation is conserved and the vortex lines must terminate at boundaries or exist in the shape of closed loops. In the zero temperature limit, a point on a vortex line will travel accordingly to the velocity field that is generated at that point by the other parts of the vortex line, provided that the vortex line is not straight (an isolated straight vortex does not move). The velocity can also be generated by any other vortex lines in the fluid, a phenomenon also present in classical fluids. A simple example of this is a vortex ring (a torus-shaped vortex) which moves at a self-induced velocity $$v_R$$ inversely proportional to the radius of the ring $$R$$, where $$R >> a_0$$. The whole ring moves at a velocity

$$v_R = \frac{\kappa}{4\pi R}\left[\ln{\left(\frac{8R}{a_0}\right)} - \frac{1}{2}\right]$$

Kelvin waves and vortex reconnections
Vortices in quantum fluids support Kelvin waves, which are helical perturbations of a vortex line away from its straight configuration that rotate at an angular velocity $$\omega$$, with

$$\omega \approx \frac{\kappa k^2}{4 \pi}\ln{\left(\frac{1}{ka_0}\right)}$$

Here $$k=2\pi/\lambda$$ where $$\lambda$$ is the wavelength and $$k$$ is the wavevector.

Travelling vortices in quantum fluids can interact with each other, resulting in reconnections of vortex lines and ultimately changing the topology of the vortex configuration when they collide as suggested by Richard Feynman. At non-zero temperatures the vortex lines scatter thermal excitations, which creates a friction force with the normal fluid component (thermal cloud for atomic condensates). This phenomenon leads to the dissipation of kinetic energy. For example, vortex rings will shrink, and Kelvin waves will decrease in amplitude.

Vortex lattice
Vortex lattices are laminar (ordered) configurations of vortex lines that can be created by rotating the system. For a cylindrical vessel of radius $$R$$, a condition can be derived for the formation of a vortex lattice by minimising the expression $$F' = F - \mathbf{L}_v\cdot\mathbf{\Omega}$$, where $$F$$ is the free energy, $$\mathbf{L}_v$$ is the angular momentum of the fluid and $$\mathbf{\Omega}$$ is the rotation, with magnitude $$\Omega$$ and axial direction. The critical velocity for the appearance of a vortex lattice is then $$\Omega^1_c = \frac{\kappa}{R^2}\ln{\left(\frac{R}{a_0}\right)}$$.

Exceeding this velocity allows for a vortex to form in the fluid. States with more vortices can be formed by increasing the rotation further, past the next critical velocities $$\Omega^2_c,\,\Omega^3_c, \cdots$$. The vortices arrange themselves into ordered configurations that are called vortex lattices.

Two fluid nature
At non-zero temperature $$T$$, thermal effects must be taken into account. For atomic gases at non-zero temperatures, a fraction of the atoms are not part of the condensate, but rather form a rarefied (large free mean path) thermal cloud that co-exist with the condensate (which, in the first approximation, can be identified with the superfluid component). Since helium is a liquid, not a dilute gas like atomic condensates, there is a much stronger interaction between atoms, and the condensate is only a part of the superfluid component. Thermal excitations (consisting of phonons and rotons) form a viscous fluid component (very short free mean path, analogous to classical viscous fluid governed by the Navier-Stokes equation), called the normal fluid which coexists with the superfluid component. This forms the basis of Tisza's and Landau's two-fluid theory describing helium II as the mixture of co-penetrating superfluid and normal fluid components, with a total density dictated by the equation $$\rho = \rho_n + \rho_s $$. The table displays the key properties of the superfluid and normal fluid components:. The relative proportions of the two components change with temperature, from an all normal fluid flow at the transition temperature $$T_c $$ ($$\rho_n/\rho \rightarrow 1 $$ and $$\rho_s/\rho \rightarrow 0 $$), to a complete superfluid flow in the zero temperature limit ($$\rho_s/\rho \rightarrow 1 $$ and $$\rho_n/\rho \rightarrow 0 $$). At small velocities, the two-fluid equations are

$$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho_s \mathbf{v}_s + \rho_n \mathbf{v}_n) = 0$$

$$ \frac{\partial(\rho S)}{\partial t} + \nabla \cdot (\rho S \mathbf{v}_n) = 0$$

$$ \rho_s\left[ \frac{\partial \mathbf{v}_s}{\partial t} + (\mathbf{v}_s \cdot \nabla)\mathbf{v}_s\right] = -\frac{\rho_s}{\rho}\nabla P + \rho_s S \nabla T$$

$$ \rho_n\left[ \frac{\partial \mathbf{v}_n}{\partial t} + (\mathbf{v}_n \cdot \nabla)\mathbf{v}_n\right] = -\frac{\rho_n}{\rho}\nabla P - \rho_s S \nabla T + \mu \nabla^2 \mathbf{v}_n$$

where here $$P $$ is the pressure, $$S $$ is the entropy per unit mass and $$\mu $$ is the viscosity of the normal fluid component as given by the table above. The first of these equations can be identified as being the conservation of mass equation, while the second equation can be identified as the conservation of entropy. The results of these equations give rise to the phenomena of second sound and thermal counterflow. At large velocities the superfluid becomes turbulent and vortex lines appear; at even larger velocities both normal fluid and superfluid become turbulent.

Classical vs quantum turbulence
Experiments and numerical solutions show that quantum turbulence is an apparently random tangle of vortex lines inside a quantum fluid. The study of quantum turbulence aims to explore two main questions:

 * 1) Are vortex tangles really random, or do they contain some characteristic properties or organised structures ?
 * 2) How does quantum turbulence compare with classical turbulence?

To understand quantum turbulence it is useful to make connection with the turbulence of classical fluids. The turbulence of classical fluids is an everyday phenomenon, which can be readily observed in the flow of a stream or river as was first done by Leonardo da Vinci in his famous sketches. When turning on a water tap, one notices that at first the water flows out in a regular fashion (called laminar flow), but if the tap is turned up to higher flow rates, the flow becomes decorated with irregular bulges, unpredictably splitting into multiple strands as it spatters out in an ever-changing torrent, known as turbulent flow. Leonardo da Vinci first observed and noted in his private notebooks that turbulent flows of classical fluids include areas of circulating fluid called vortices (or eddies).

The simplest case of classical turbulence is that of homogeneous isotropic turbulence (HIT) held in a statistical steady state. Such turbulence can be created inside of a wind tunnel, for example a channel with air flow propelled by a fan from one side to the other. It is often equipped with a mesh to create a turbulent flow of air. A statistically steady state ensures that the main properties of the flow stabilises even though they fluctuate locally. Due to presence of viscosity, without the continuous supply of energy the turbulence of the flow will decay because of frictional forces. In the wind tunnel, energy is consistently provided by the fan. It is useful to introduce the concept of energy distribution over the length scales, the wavevector $$\mathbf{k} = (k_x,k_y,k_z)$$, and the wavenumber $$k = |\mathbf{k}|$$. In one dimension, the wavenumber can be related to the wavelength simply using $$k = 2\pi/\lambda$$. The total energy $$E_{tot}$$ per unit mass is given by

$$E_{tot} = \frac{1}{\mathcal{V}}\int_{\mathcal{V}}\frac{\mathbf{v}^2}{2}\, \mathrm{d}^3\mathbf{r} = \int_{0}^{\infty}E(k)\, \mathrm{d}k$$

where $$E(k)$$ is the energy spectrum, essentially representing the distribution of turbulent kinetic energy over the wavenumbers. The notion of an energy cascade, where an energy transfer takes place from large scale vortices to smaller scale vortices, which eventually lead to viscous dissipation, was memorably noted by Lewis Fry Richardson. Dissipation occurs at the dissipation length scales $$\eta$$ (termed the Kolmogorov length scale), where $$\eta = (\nu^3/\varepsilon)^{1/4}$$ where $$\nu$$ is the kinematic viscosity. By the pioneering work of Andrey Kolmogorov, the energy spectrum was found to take the form

$$E(k) = C \varepsilon^{\frac{2}{3}}k^{-\frac{5}{3}}$$

where $$\varepsilon$$ is the energy dissipation rate per unit volume $$-\mathrm{d}E/\mathrm{d}t$$. The constant $$C$$ is a dimensionless constant, that takes the value $$C \approx 1.5$$. In k-space the value associated to the Kolmogorov length scale is the Kolmogorov wavenumber $$k_{\eta}$$, where viscous dissipation occurs.

Kolmogorov cascade in quantum fluids
For temperatures low enough for quantum mechanical effects to govern the fluid, quantum turbulence is a seemingly chaotic tangle of vortex lines with a highly knotted topology, which move each other and reconnect when they collide. In a pure superfluid, there is no normal component to carry the entropy of the system and therefore the fluid flows without viscosity, resulting in the lack of a dissipation scale $$\eta$$. Analogously to classical fluids, a quantum length scale $$\ell$$ (and the corresponding value in k-space $$k_{\ell}$$) can be introduced by replacing the kinematic viscosity in the Kolmogorov length scale with the quantum of circulation $$\kappa$$. For scales larger than $$\ell$$, a small polarisation of the vortex lines allows the stretching required to sustain a Kolmogorov energy cascade.

Experiments have been performed in superfluid Helium II to create turbulence, that behave according to the Kolmogorov cascade. One such example of this is the case of two counter-rotating propellers, where both above and below the critical temperature $$T_c$$ a Kolmogorov energy spectrum was observed that is indistinguishable from those observed in the turbulence of classical fluids. For higher temperatures, the existence of the normal fluid component leads to the presence of viscous forces and eventual heat dissipation which warms the system. As a consequence of this friction the vortices become smoother, and the Kelvin waves that arise due to vortex reconnections are smoother than in low-temperature quantum turbulence. Kolmogorov turbulence arises in quantum fluids for energy input at large length scales, where the energy spectrum follows $$k^{-5/3}$$ in the inertial range $$k_D < k < k_{\ell}$$. For length scales smaller than $$\ell$$, instead the energy spectrum follows a $$k^{-1}$$ regime.

For temperatures in the zero limit, the undamped Kelvin waves result in more kinks appearing in the shapes of the vortices. For large length scales the quantum turbulence manifests as a Kolmogorov energy cascade (numerical simulations using the Gross-Pitaevskii equation and the vortex-filament model confirmed this effect ), with the energy spectrum following $$k^{-5/3}$$. Lacking thermal dissipation, it is intuitive to assume that quantum turbulence in the low temperature limit does not decay as it would for higher temperatures, however experimental evidence showed that this was not the case: quantum turbulence decays even at very low temperatures. The Kelvin waves interact and create shorter Kelvin waves, until they are short enough that emission of sound (phonons), which results in the conversion of kinetic energy into heat, thus dissipation of energy. This process which shifts energy to smaller and smaller length scales at wavenumbers larger than $$k_{\ell}$$ is called the  Kelvin wave cascade and proceeds on individual vortices. Low temperature quantum turbulence should thus consist of a double cascade: a Kolmogorov regime (a cascade of eddies) in the inertial range $$k_D < k < k_{\ell}$$, followed by a bottle-neck plateau, followed by the Kelvin wave cascade (a cascade of waves) that obeys the same $$k^{-5/3}$$ law but with different physical origin. This is at current consensus, but it must be stressed that it arises from theory and numerical simulations only: there is currently no direct experimental evidence for the Kelvin wave cascade due to the difficulty of observing and measuring at such small length scales.

Vinen turbulence
Vinen turbulence can be generated in a quantum fluid by the injection of vortex rings into the system, which has been observed both numerically and experimentally. It has been observed also in numerical simulations of turbulent Helium II driven by a small heat flux and in numerical simulations of trapped atomic Bose-Einstein condensates; it has been found even in numerical studies of superfluid models of the early universe. Unlike the Kolmogorov regime which appears to have a classical counterpart, Vinen turbulence has not been identified in classical turbulence..

Vinen turbulence occurs for very low energy inputs into the system, which prevents the formation of the large scale partially polarised structures that are prevalent in Kolmogorov turbulence, as is shown in Fig 9a. The partial polarization contributes strongly to the amount of non-local interactions between the vortex lines, which can be seen in the figure. In stark contrast, Fig 9b displays the Vinen turbulence regime, where there is very little non-local interaction. The energy spectrum of Vinen turbulence peaks at the intermediate scales around $$k_{\ell}$$, rather than at large length scales $$k_{D}$$. From Fig 10, it can be seen that for small length scales the turbulence follows the typical $$k^{-1}$$ behaviour of an isolated vortex. As a result of these properties Vinen turbulence appears as an almost completely random flow with a very weak or negligible energy cascade.

Decay of quantum turbulence
Stemming from the different signatures, Kolmogorov and Vinen turbulence follow power laws relating to their temporal decay. For the Kolmogorov regime, after removing the forcing which sustains the turbulence in a statistical steady-state, a decay of $$E(t) \sim t^{-2}$$ for the energy and $$L(t) \sim t^{-3/2}$$ for the vortex line density (defined as the vortex length per unit volume) are observed. Vinen turbulence decays temporally at a slower rate than Kolmogorov turbulence: the energy decays as $$E(t) \sim t^{-1}$$ and the vortex line density as $$L(t) \sim t^{-1}$$.

Turbulence in atomic condensates
Computer simulations have played a particularly important role in the development of the theoretical understanding of quantum turbulence

Turbulence in atomic condensates has only been studied very recently meaning that there is less information available. Turbulent atomic condensates contain a much smaller number of vortices compared to turbulence in helium. Because of the small size of typical atomic condensates, there is not a large length scale separation between the system size and the inter-vortex size, and therefore k-space is restricted. Numerical simulations suggest that turbulence is more likely to appear in the Vinen regime. Experiments performed in Cambridge have also found the emergence of wave turbulence scaling appearing.

Physical generation of quantum turbulence
There are a plethora of methods that can be used to generate a vortex tangle (visualised in fig 11) in the laboratory. Here they are listed by the quantum fluid that they can be generated in.

QT in helium II

 * Suddenly towing a grid in the sample of fluid at rest
 * Moving the fluid along pipes or channels using bellows or pumps, creating a superfluid wind tunnel (the TOUPIE experiment in Grenoble )
 * Rotating one or two propellers inside a container; the configuration of two counter-rotating propellers is called the "von Karman flow" (e.g. the SHREK experiment in Grenoble)
 * Creating shockwaves and cavitation by locally focusing ultrasound (this allows for the generation of quantum turbulence away from the boundaries)
 * Oscillating/vibrating forks or wires
 * Applying a heat flux (also termed the "thermal counterflow" ): the prototype experiment is a channel which is open to a helium bath at one end and the opposite is closed and has a resistor. An electric current is passed through the resistor and generates ohmic heat; the heat is carried away from the heater towards the bath by the normal fluid component, while the superfluid moves towards the heater so that the net mass flux is zero as the channel is closed. A relative velocity $$v_{ns} = |v_n - v_s|$$ (counterflow) of the two fluid components is set up in this way which is proportional to the applied heat. Above a small critical value of the counterflow velocity, a turbulent vortex tangle is generated.
 * Injecting vortex rings (rings are generated by injecting electrons which form a small bubble of about 16 Angstroms in size that are accelerated by an electric field, until, upon exceeding the critical velocity, the vortex ring is nucleated)

QT in 3He-B and atomic condensates
In 3He-B, quantum turbulence can be generated by the vibration of wires. For atomic condensates, quantum turbulence can be generated by shaking or oscillating the trap which confines the BEC and by phase imprinting the quantum vortices.

Detection of quantum turbulence
In classical turbulence, one usually measures the velocity, either at a fixed position against time (typical of physical experiments) or at the same time at many positions (typical of numerical simulations). Quantum turbulence is characterised by a disordered tangle of discrete (individual) vortex lines.

In helium II techniques exists to measure the vortex line density $$L$$ (the length of vortex lines per unit volume based on detecting the second sound attenuation. The average distance between vortex lines, $$\ell$$, can be found in terms of the vortex line density as $$\ell = 1/\sqrt{L}$$.

Detection in helium II

 * Measuring the attenuation of second sound waves
 * Measuring temperature or pressure gradients
 * Measuring ions trapped in the vortices
 * Using tracer particles (small glass or plastic spheres/solid hydrogen snowballs) of size of the order of a micron, and then imaging them using lasers. Techniques that can be used are PIV (particle image velocimetry) or PTV (particle tracking velocimetry). Most recently, excimer helium molecules have been used
 * Using oscillating forks
 * Using cantilevers
 * Using cryogenic hot wires

Detection in 3He-B and atomic condensates
Quantum turbulence can be detected in 3He-B in two ways: nuclear magnetic resonance (NMR) and by Andreev scattering of thermal quasiparticles. For atomic condensates, it is typical that the condensate must be expanded (by switching off the trapping potential) so that is sufficiently large for an image to be taken. This procedure has a disadvantage as it leads to the condensate being destroyed. The outcome leads to a 2-dimensional image which allows for the study of 2-dimensional quantum turbulence, but imposes a constraint studying 3-dimensional quantum turbulence using this method. Individual quantum vortices have been observed in 3-dimensions, moving and reconnecting using a technique which extracts small fractions of the condensate at a time, allowing for the observation of a time sequence of the same vortex configuration.

title: Quantum turbulence


text:
Hongcheon County

Hongcheon (Hongcheon-gun) is a county and city in Gangwon Province, South Korea. The city lies on the northern bank of the Hongcheon River, southeast of Chuncheon. The terrain of the county is mainly mountainous and contains hot springs in the Hongcheong River valley. The county produces ginseng, maize, and vegetables. As of 2012 the county had a population of 70,401 people (including foreigners) in 29,894 households.

History
Hongcheon's historic name was BulRyeokChunHyeon during the rule of Goguryeo. King Gyeongduk of Silla changed the name to YoungHyeon of Sak-Ju, In 1043, it was named Hongcheon during the reign of King In-Jong. Following the independence of the country on 15 August 1945, many administrative changes were made, affecting the jurisdiction of various units under its control.

Korean Federation for Environmental Movement (KFEM) opened an educational facility in Hongcheon in 1995.

Hongcheon County uses an emblem to represent itself as the “County of Culture and Tourism.” The emblem, a simple depiction of a leaf and dew, symbolizes a clean environment, patriotism, and love of one's hometown.

Geography
Hongcheon is located in Gangwon Province. The county, located in the middle of the peninsula, is the largest in Korea with an area of about 1818 km2 and is reported to be three times the size of Seoul. It is bounded by Chuncheon city on the north, the Inje County and Yangyang County to the northeast, by Gapyeong County of Gyeonggi Province to the west, by Hoengseong and Pyeongchang counties to the north, and Pyeongchang county to the south. The county is a middle-level mountainous region, an offshoot of the Baegdu mountain range. The eastern end of the mountain range, where Myeonggae village and Nae township are located, runs in an east–west direction for a length of 93.1 km The western part of the mountain range which has Dongmak village and  Seotown extends over a length of 39.4 km in an east–west direction. While the southern part of the hill range covers the Sidong village and the Nam town, the northern part has Jangnam village and Duchon township. The Palbong Mountain (327.4 m) has scenic views of the course of the Hongcheon River; the river forms white sandy banks. The Hongcheon river system which rises the Taebaek mountain watershed, joins the Hongcheon River which forms a small plain area at Seorak-myeon, Gapyeong-gun, Gyeonggi-do. It then flows in to the Cheongpyeong Dam in Gyeonggi Province. Gachilbong Sambong Spring is located in Gwangwon-ri, Nae-myeon and originates from the Palbong Mountain; Garyeong Waterfall is located in  Waya-ri, Naechon-myeon at the foot of Mt. Baekam (1,099m) which drops through rapids over a height of 50 m. The area is known for medicinal herbs, wild flowers and mountain birds. The bus terminal provides connections with Gimpo and Incheon airports. A bus ride from Dong Seoul Bus Terminal to Hongcheon is approximately one hour. Naerincheon, a small river, crosses Hongcheon and is the only one in the country which flows north. Nearby is the Daemyung Vivaldi Park Ski World, a ski resort (with 13 ski runs, each with a music theme) and water park called OceanWorld in the mountains. Local football club Hongcheon Idu FC play their games at Hongcheon Public Park.

One of the attractions of Hongcheon is Alpaca World. Located in the forest of Pungcheon-ri, Hwachon-myeon, Hongcheon-gun, this place is designed to provide communication and experience with animals. It is popular with people because they can meet various animals as well as alpacas.

Climate
The climate of Hongcheon County is a humid continental climate (Köppen: Dwa). Climatic conditions are considered extreme as its location is away from the sea. The minimum temperature recorded is –28.1 °C in January. The climatic data is given in the table below. On 1 August 2018, Hongcheon County recorded a temperature of 41.0 C, which is the highest temperature to have ever been recorded in South Korea.

Flora and fauna
Daisy fleabane, a native plant of North America, was inadvertently introduced to Korea in the 1990s and is found in Hongcheon County also. A strange spot on its leaf was found in daisy fleabanes in June 2011. This was tested in the laboratory and found to be caused by a pest identified as Septoria erigerontis.

Rice cultivation
The dominant crop grown in the country is called the Sura rice. The county, with its widely varying temperatures between day and night times, and with clean and un polluted water, provides an ideal environmental condition ideal to grow this crop which is considered to be of very high quality with cooked rice retaining its taste for much longer than other species of rice. It is grown in 38% of the land under cultivation in the county. It is also disease resistant and can be stored for a long time.

Culture
The county is noted for its beef production with its climatic conditions favorable for raising male calves (bred by artificial insemination) which are bred for their exquisite flavor and texture. They are fed alcohol-fermented feed which is reported to reduce the cholesterol content and give flavor to the beef. The calves are raised for minimum of 27 months.

Cultural festivals celebrated in Hongcheon include the Chal Oksusu (Corn) Festival, Mugunghwa (Hibiscus) Festival, and Ginseng Festival. The Mugunghwa Festival honors the memory of Namgung Eok, a statesman and journalist from Hongcheon. The Hongcheon Ginseng Festival, first held in 2003, is held annually in October to celebrate the medicinal properties of ginseng, considered a miracle medicine in Korea. Traditional Taekkyeon and Dudrak performances are part of the festivities. Previously, the county celebrated Choi Seung-hee Dance Festival in honor of a dancer who developed modern forms of traditional folk dances, but because she spent the latter years of her life in North Korea, she is no longer celebrated.

Symbol Mark
Among the five colors of Mugunghwa petals, blue symbolizes the spirit of the county people, green symbolizes the unpolluted natural environment. Pink symbolizes the fragrance of flowers. Yellow symbolizes the peace of Hongcheon. Red symbolizes the passion of Hongcheon. The center of the flower represents the cohesive power of the Hongcheon people and the green water droplets in the lower part represent a clean earth.

Character
The name of the character is Mugung-gi (literally Eternal-flag) - a symbol of Hongcheon with MugungHwa (literally Eternal Flower) central to its design incorporating other elements.

The water droplets representing the nature of Hongcheon were expressed to the face. The high-tech antenna on the right expressed its meaning as a character leading the information age.

Notable sites
The Korean peninsula has been subject to archaeological excavations for establishing obsidian (a naturally occurring volcanic glass formed as an extrusive igneous rock) at many Paleolithic sites. One such site excavated is located in Hongcheon county, known as Hahwageri III. Several samples collected from the site have been subject to carbon dating. In the first cultural layer, out of the two layers established during the excavations, the finds consists of microliths of obsidian and quartz crystal which are embedded in microblade cores. In addition, arrowheads, cores, anvils, hammerstones and pecking tools have also been recovered, apart from a large cache of obsidian artifacts.

The Suta Temple of Mt Gongjak is a Buddhist shrine at the foot of the Gongjak Mountain which appears like a peacock spreading its wings and its flora consists of azalea and ancient pine trees. It is a national monument that was built in 708, in stone, by the Buddhist priest WonHyo during the reign of King SeongDeok, the 33rd King of Silla.

Notable people
It is the birthplace of Lee Young-Pyo, who gained global recognition as the legendary leftback of the 2002 South Korea national football team which reached the semifinal of 2002 World Cup, and went on to play for various top tier teams in Europe such as Tottenham Hotspur and PSV Eindhoven to great acclaim.

title: Hongcheon County


text:
Lossiemouth

Lossiemouth (Inbhir Losaidh) is a town in Moray, Scotland. Originally the port belonging to Elgin, it became an important fishing town. Although there has been over 1,000 years of settlement in the area, the present day town was formed over the past 250 years and consists of four separate communities that eventually merged into one. From 1890 to 1975, it was a police burgh as Lossiemouth and Branderburgh.

Stotfield, the first significant settlement (discounting Kinneddar which has now disappeared), lies to the north west of the town. Next was the Seatown – a small area between the river and the canal inholding of 52 houses, 51 of which are the historic fisher cottages. When the new harbour was built on the River Lossie, the 18th-century planned town of Lossiemouth, built on a grid system, was established on the low ground below the Coulard Hill. Branderburgh formed the final development during the 19th century. This part of the town developed entirely as a result of the new harbour with its two basins, and eventually covered the entire Coulard Hill; it has an impressive profile when seen from a distance.

Roman to Medieval

Although the Romans never conquered the peoples of the north of Scotland, they made several journeys to the Moray Firth coast. Suspected Roman forts have been discovered at Thomshill, near Elgin and at Easter Galcantray, Cawdor, Nairnshire as well as a suspected marching camp at Wester Alves, Moray. The Greco-Roman astronomer and geographer Ptolemy (c. 90 – c. 168), describes in chapter two of his Geographa entitled Albion Island of Britannia the mouth of the River Lossie as ostium Loxa Fluvius. Settlement in this area has a long history. St Gervadius, a Celtic hermit, inhabited a cave overlooking the entrance to the sea loch, Loch Spynie. In his time, the River Lossie entered the loch further to the south, near Inchbroom. The rocky promontory is recorded in the Chartulary of Moray as Holyman's Head and it is said that Gervadius (St Gerardine as he became known in later times) would walk around the headland with a flaming torch to warn ships away from the dangerous rocks. Even today the Halliman Skerries retain the reference to St Gervadius. He died in 934 and his cave was a place of pilgrimage right up to the 16th century. The cave was eventually quarried out.

The settlement at the river mouth is significant particularly in its relationship with the royal burgh of Elgin. An argument between Alexander Bur, Bishop of Moray and John Dunbar, 4th Earl of Moray was documented in 1383 regarding the "ownership" of the port of "Losey". This document mentions that Losey was commonly known to fall within the limits of the episcopal estates. The Bishop's description of the port suggests that it was well downstream from his fishing station at Spynie. So it seems likely that Losey was not merely a fishing station but also a trading port that the Elgin burgesses used as a counterbalance to the Royal Burgh of Forres's trading port of Findhorn. The dispute with the Earl of Moray went further. That same year of 1383, the Earl wrote to the Elgin burgesses offering them the use of his port at the mouth of the River Spey with no duties, in an attempt to take trade from the Bishop. The port and fishery was mentioned again in 1551.

The loch and the river became separated around 1600. A succession of storms built banks of sand and boulders that eventually closed off the sea entrance. It is documented that in 1609, to avoid flooding, the post-Reformation Protestant Bishop Alexander Douglas took steps to exclude the River Lossie from the loch. Evidence of a sudden and apparently artificial right-angled bend between Caysbriggs and Inchbroom may indicate the location of this diversion.

Modern Lossiemouth has its origins in five separate communities that in time grew into one. These were Kinneddar, Stotfield, Seatown, Lossiemouth, and finally Branderburgh; the most ancient of these are Kinneddar and Stotfield.

Kinneddar

Kinneddar (Gaelic: Cinn Eadar) has now disappeared as a ferm toun, however an old farmhouse still retains its name and is probably its location. A Pictish settlement occupied the area and large numbers of carved stones, now held in Elgin Museum, were found. These stones date the settlement to around the 8th or 9th century. Pictish crosses were found in or near the cemetery and indicate the presence of a Christian establishment. Early documented references to the settlement refer to it as Kenedor dating it to the 10th century; it may, of course, have been a continuation of the original Pictish religious community. Saint Gervadius (Gerardine) is referred to as "Gervadius of Kenedor" and may have been part of this community, establishing his cell in the cave just to the northeast.

Bishop Richard is known to have resided at Kinneddar and for that period, it became the cathedral church of the diocese. However maps dating from the early 16th century clearly show this farming community as King Edward. It is unlikely, though, that this community took its name from King Edward I of England, The Hammer of the Scots, even though Edward travelled twice to this area to demonstrate his grip over the country; the most likely explanation is that the early cartographers took the local pronunciation of Kinneddar as King Edward and recorded it as such. He is known to have stayed in Elgin for four days in late July 1296 and it was during this sojourn into Scotland that he removed the Stone of Scone (Stone of Destiny) from Scone Palace and had it placed in a wooden chair at Westminster Abbey. He again stayed in Elgin for two days in September 1303 and then camped at Kinloss Abbey from 13 September to 4 October.

At that time the castle at Kinneddar, along with those at Elgin and Duffus, was left under the control of English garrisons. In 1308, Robert the Bruce, taking advantage of King Edward II's preoccupation with matters in England and France, started capturing and usually destroying castles that were either English garrisoned or controlled throughout Scotland. Joined by an army provided by David de Moravia, the Bishop of Moray, Bruce burned the castles of Inverness and Nairn before seizing and burning Kinneddar castle. He attacked Elgin castle only to be twice repulsed before finally succeeding. King Edward had the Bishop ex-communicated causing him to flee to Norway only to return after Edward's death.

Kinneddar village was still sizeable in the early 19th century but dwindled away with the building of the new Lossiemouth, just to the east.

Stotfield

The early maps, some dating back to the early 16th century, clearly show Stotfield; some maps name the settlement as Stotfold or Stodfauld. The name Stotfold comes from the Middle Scots meaning, 'horse fold'. King David I introduced settlers from other parts of the kingdom as a way of reducing the powers of the lords who had ruled large territories as independent provinces. Indeed, King David put down a rebellion by the Mormaer of Moray in 1130 and it is possible that Stotfield dated from shortly after this event. The English-speaking inhabitants of the Lothians would most likely have been the chosen settlers. It is notable that the people inhabiting the Lothians were Angles (formally part of the Kingdom of Northumbria).

In the Middle Ages, Stotfield was primarily a farm hamlet with small-scale fishing. The fishing gradually became more important and the population specialised into farm workers and fishermen. However, the religious strictures introduced in the Reformation impacted the community, especially the fishers. The minutes of the Kinneddar Parish Kirk Session clearly show that the ancient superstitions were at least as important to them as observance of church niceties: "'17 Aprilis 1670 After Sermon the Session Assembling &c. The said day the fishers of Stotefold & Cousea being remitted from þe Presbetry to this Church discipline for satisfaction of þr (their) great & gross scandall & Idolatrous custome in burning torches on þe new years even The Presbetry having ordained þ' (that) those psons mor in accession in this transgression þn (than) oþrs (others) satisfy þe discipline in Sacco And oyra according to the arbitrement of þe Sessione. The Session do þrfore (therefore) ordain John Edward in Stotefold to satisfy in Sacco on day & to pay 20s James Jafray in Cousea to satisfy in the Joges two dayes, Wm Innes Wm Hesbein Thomas Edward & John Thome all of þm (them) to testify þr (their) Repentance by standing at þe pillar And ilk ane of þm to pay 20s. Alexr Innes owner of þe Boats of Stotefold, Wm Young owner of þe boats of Cousea each of þm are ordained to pay 4 libs (pounds). In regard that they had not restrained this abuse Conform to þr (their) engagement before þe Presbetry in Ano 66 (year 1666) The fors (four) psons (persons) all of þ snd Compeiring þr sentence being intimated unto þm they accepting & submitting to disciplin were sharpely rebuked exhorted to serious Repentance & enjoyned to satisfy conform to þe ordinance The next Lords day.'"

But the practices continued, and 35 years later the minutes from the session records stated: "'23 Dec 1705 Also after sermon þe min1' (the minister) did guard þe Seamen to beware of þe old Heathenish superstitious practice of carrieing of lighted Clevies or torches about þr boats on new years even certifieing all that should be found any manner of way to concurr with or contribute to þe said work—should be put in þe hands of þe civill magistrate.'"

This is interesting because it shows that the power to fine parishioners had by then been removed and put in the hands of magistrates.

Parish records from Duffus Kirk show that similar events were happening at Brughsea (Burghead). It is apparent, therefore, that Clavie burning was carried out in the three fisher towns of Brughsea, Causie (Covesea) and Stotefold (Stotfield). It is unlikely that this practice would have been restricted to the three Morayshire locations and likely that it would have been more widespread. Burghead still burns a ceremonial clavie on the eve of the old (Julian) new year but it is no longer associated with fishing boats. A puzzling date for the modern ceremony as the 17th-century ones were held on 31 December.

Stotfield fishing disaster
The Stotfield fishing disaster struck on 25 December 1806 and every able-bodied male in the village perished in the storm; this had an enormous effect on the Stotfield community. The folk memory of it is still retained among the fishermen of Lossiemouth.

Seatown

The Seatown was established at the end of the 17th century when the old port at Spynie became landlocked. A succession of storms had built up large shingle banks to block the outlet of Loch Spynie to the sea. The merchants of Elgin decided that a new harbour that could berth larger trading vessels at the river mouth was required. The fishermen did not use the new pier, however, but continued to sail their boats up to the beach at the Seatown. Seatown is called The Toonie by its inhabitants and sometimes referred to as the Dogwall. This was a reference to dog-skins that were dried here before being turned into floats for nets.

Lossiemouth

In 1685, the Elgin burgh council called upon a German engineer, Peter Brauss, to look at the viability of providing a harbour at the mouth of the River Lossie; he decided that a harbour could be established. The first efforts at the beginning of the 18th century looked to have failed but by 1764, the new jetty had been built at a cost of £1200.

At the time that the new river mouth harbour was being constructed, so too was a more planned development laid out in streets running parallel and right angles to each other. An open square with a cross separated the first settlement from the new. The fishers occupied the houses at the Seatown and the builders, craftsmen and merchants in the new Lossiemouth. Later, a canal cut to drain Loch Spynie, would present a physical barrier to the two communities and entered the River Lossie in this area.

Branderburgh

By the early 19th century, the river harbour was busy but its long-term future was unsustainable and meant that a new solution was sought. In 1834, a Stotfield and Lossiemouth Harbour Company was formed to look into building a new harbour at Stotfield Point. That same year, The Inverness Courier carried the following: "'A paragraph is quoted from an Elgin paper under the heading 'unexampled economy worthy of imitation.' The two senior bailies of the burgh went on behalf of the town to Lossiemouth to meet the gentlemen appointed to stake off the ground for a proposed new harbour. The worthy Magistrates walked the whole distance, five miles out and five miles home, and only spent one shilling! This expenditure consisted of sixpence for whisky and the other sixpence to the waiter.'"

The construction of the new harbour was carried out between 1837 and 1839 but initially in a relatively small form. The beginning of the building process was marked by a ceremony and reported in the Inverness Courier as follows, "'The ceremony of laying the foundation stone of the inner basin of the new harbour at Stotfield Point, Lossiemouth, took place on the 15th inst [June]. The stone was laid by Lieut. Colonel James Brander of Pitgaveny, the proprietor of the site, with the assistance of the Trinity Lodge of Freemasons, and in presence of the Chairman and shareholders of the Harbour Company, and representatives of the burgh of Elgin.'" This was the beginning of the final phase of building that was to become Branderburgh. However, by 1858, the basin had been enlarged further and deepened to 16 ft at spring tides. This encouraged many fishing families from up and down the coast to move to the town. The harbour as well as having a large herring fleet by now, also shared the available space with trading ships. This prompted the now renamed Elgin and Lossiemouth Harbour Company to build a new second basin at a cost of £18,000. This basin was intended solely for fishing boats and opened in 1860.

Branderburgh, with its characteristic wide streets, continued to push its boundaries westward and by the early 20th century finally joined with Stotfield. A substantial amount of sandstone was quarried from the east side of the town to accommodate this rapid house building project. After Lossiemouth and Branderburgh became a police burgh in 1863, the town evolved to become known as Lossiemouth. Lossiemouth Town Hall was completed in 1885.

Fishing boats
The boats used at Stotfield, Seatown and finally Branderburgh were the same as those found across the entire Scottish east coast fishery. Chronologically, these were the two masted luggers, the Skaffies, Fifies and Zulus; then the powered Steam Drifters and Seine Netters.
 * The Skaffie appeared at the beginning of the 19th century. These boats were initially small so that they could be easily beached, but later versions were heavier when large harbours became prevalent. Their stems were rounded and had raked sterns.
 * The Fifie was the predominant fishing boat on the east coast from the 1850s until the mid-1880s. The Fifie's main features were the vertical stem and stern. Fifies built from 1860 onwards were all decked and from the 1870s onwards the bigger boats were built with carvel planking, i.e. the planks were laid edge to edge instead of the overlapping clinker style of previous boats. Some boats were built up to about 70 ft in length and were very fast.
 * The Zulu took its name from the Zulu war that was raging in South Africa at the time. Lossiemouth fisherman William 'Dad' Campbell was the first to introduce this form of fishing boat. His boat, the Nonesuch, had the characteristic vertical stem and steeply raking stern. The Zulu Boats rapidly became very popular in Lossiemouth and then along the whole of the east coast. Because these boats were ultimately very big and fast, they could reach the fishing grounds quickly and return with the catch equally fast.
 * The Steam Drifters were so called because just like the Fifies and Zulus, they used drift nets. They were large boats, usually 80 – in length with a beam of around 20 ft. Steam drifters had many advantages. They were usually about 20 ft longer than the sailing vessels, so they could carry more nets and catch more fish. This was important because the market was growing quickly at the beginning of the 20th century. They could travel faster and further and with greater freedom from weather, wind and tide. Because less time was spent travelling to and from the fishing grounds, more time could be spent fishing. However they did have disadvantages: they were expensive to build and run and as the herring fishery declined they became too expensive to operate.
 * The Seine Netters initially were converted Fifies and Zulus. From 1906, petrol and paraffin engines began to be installed, initially for auxiliary power. However, as more powerful engines became available, sails (apart from the mizzen sail) were dispensed with. Danish seine net boats were landing huge quantities of plaice and other white fish at English east coast ports. Lossiemouth fishermen noted this and a few decided to use the seine net. It was obvious that this would be successful, but they were still hampered by the design and cost of the majority steam boats. John Campbell, nephew of William Campbell who designed the first Zulu boat, saw that a new design was needed to accommodate the large amounts of white fish that could be caught. His boat, the Marigold, did very well and over a short period the entire fleet (the first in Scotland) converted to the seine net.

Morayshire Railway
The Morayshire Railway was officially opened at ceremonies in Elgin and Branderburgh on 10 August 1852, the steam engines having been delivered to Lossie by sea. It was the first railway north of Aberdeen. Initially it went only the 5+1/2 mi between Lossie and Elgin, but it was later extended south to Craigellachie. The Lossie–Elgin section had three intermediate stops: the Rifle Range Halt, Greens of Drainie and Linksfield. The Great North of Scotland Railway took over the working of the line in 1863 and bought the company in 1881 following the Morayshire Railway's return from crippling debt back to solvency. The railway and harbour became very important to the economy of both Lossie and Moray. It was the Morayshire Railway that persuaded Col Brander, of Pitgaveny, to build the bridge from the Seatown to the east beach to encourage more day tripping in the summer months.

Timeline
Dates

Note: From early maps, Stotfield is known to exist in the 16th century (see National Library of Scotland) but is probably older. Kinneddar is referenced in the 10th century.

Geography, geology and wildlife

The town is located on the most northerly point of the south coast of the Moray Firth, at the mouth of the River Lossie. To the west of the town are a sandy beach, golf links and the Royal Air Force station, RAF Lossiemouth. Lossie Forest is a large pine forest that starts on the town's south-east boundary and the river splits it into two sections. The south side of the town is joined by the fertile plains of the Laich o' Moray.

Lossiemouth Beach is a large strip of dunes separated from the rest of the town by the River Lossie, creating a useful sheltered expanse of water. The town looks down onto this natural harbour with a plain promenade street from which the East Beach Bridge leads onto the sands. Ringed plover, grey heron, black-headed gull, oystercatcher, curlew, mallard and other waders feed under the bridge and are easy to watch from the street, and there are vast numbers of water birds in the more rural area further east.

A large part of the town is built on the Coulard Hill which consists of pale grey and yellow sandstones and with these is associated a cherty and calcareous band, known as  'the cherty rock of Stotfield' . This rock is a form of silica that contains micro-crystalline quartz. Also in the calcareous band of the Stotfield rock there is limestone with nodular masses of flint, crystals of galena (lead ore) and iron pyrites. The quarry on the east side of the town that produced the stone for the building of Branderburgh produced the largest variety and total numbers of fossil reptiles from the late Triassic Period to have been found in the UK. This was a total of eight species and 97 individuals; five of the species are unique to Lossiemouth, one of which is an early form of dinosaur. This quarry is ranked as one of Britain's most important fossil bearing locations of this period.

Demography
Source: Moray Council from 2001 Census data Lossiemouth's population in 1901 was 3904

Economy
Lossiemouth is heavily dependent on the Royal Air Force station for its employment of civilians. In 2005, RAF Lossiemouth along with its neighbour RAF Kinloss contributed £156.5 million (including civilian expenditure) to the Moray economy, of which £76.6 million was retained and spent locally. The bases are responsible for providing, directly or indirectly, 21 per cent of all employment in the area.

Before the RAF, fishing was almost the sole industrialised process in Lossiemouth, but it now contributes very little to the overall economy; in 2001, fishing accounted for 1.62% (55 individuals) of the full-time employment of Lossiemouth. However, some engineering businesses once associated with fishing still exist in the town. Unlike other sizable towns in the area, Lossiemouth has not attracted the large supermarket groups due to its proximity to Elgin, which also provides employment to a significant proportion of the working population.

Source: Highland & Islands Enterprise & Moray Council

Transport
Three roads converge on the town. The A941 connects to Elgin, while the B9103 joins the A96 (main Inverness to Aberdeen route) and the B9040 connects to Hopeman and Burghead. There is a regular bus service to and from Elgin.

The nearest railway station is at Elgin and offers services every 90 to 120 minutes to both Inverness (about 50 minutes travel time) and Aberdeen (about 90 minutes travel time), and onwards to the rest of the UK. The former Morayshire Railway line to Elgin was closed to passenger traffic in 1964 and goods in 1966. The former railway route has been turned into a footpath, and the edge of the station platform is visible in the main car park near the harbour.

There is a frequent bus service from Lossiemouth to Elgin.

Inverness Airport (36 miles / 58 km) and Aberdeen Airport (62 miles / 100 km) offer a wide range of destinations.

National governments

 * Lossiemouth is in the Moray (Westminster) constituency of the United Kingdom Parliament which returned Douglas Ross as its Member of Parliament (MP) in the last general election.
 * Lossiemouth is in the Moray constituency of the Scottish Parliament has slightly different boundaries to the UK Parliament constituency of the same name. It is represented at Holyrood by Richard Lochhead. The town is also part of the Highlands and Islands electoral region.

Local government

 * Lossiemouth is part of the Moray council area and elects its four representatives as part of the Heldon and Laich ward.

Primary

 * St Gerardine’s Primary School
 * Hythehill Primary School

Secondary

 * Lossiemouth High School is located in the south west of the town by the playing fields. Adjacent to the school is the swimming pool and community centre, which incorporates a playschool. Lossie High serves the Burghsea area, dominated by Lossiemouth itself and the villages Hopeman, Burghead, Cummingston and Duffus. The feeder primaries are Hythehill, St. Gerardine's, Hopeman and Burghead. There are over around 500–600 pupils (As of March 2020) separated into four houses; Covesea, Kinneddar, Pitgaveny and Spynie.

Further

 * Ecosse Performers College is located at 56 High Street.

Religion
The following religious denominations have places of worship in Lossiemouth:

Church of Scotland
 * St Gerardine's High, Drainie Parish Church, St Gerardines Road
 * St James’ Church, Prospect Terrace

United Free Church of Scotland
 * United Free Church, St Gerardines Road

Baptist
 * Lossiemouth Baptist Church King Street

Plymouth Brethren
 * Gospel Hall, James Street

Scottish Episcopal
 * St Margaret’s Church, Stotfield Road

Roman Catholic
 * St Columba's Church, Union Street

Culture and leisure

 * Lossiemouth Cruising Club The Marina, Pitgaveny Quay
 * Moray Golf Club, Stotfield Road. The club has two 18 hole courses.
 * The Warehouse, Pitgaveny Quay – Small intimate theatre/live music venue
 * Lossiemouth Fisheries and Community Museum, Pitgaveny Quay
 * Public Library, Town Hall Lane
 * Swimming pool, adjacent to Lossiemouth High School
 * Covesea Lighthouse off B9040 now owned by Covesea Lighthouse Community Co. Ltd.
 * RAF and FAA Heritage Centre at Covesea Lighthouse

Sport

The town's main football club is Lossiemouth F.C., nicknamed "The Coasters", who play in the Highland Football League. The club play their home games at Grant Park. They have won several trophies in recent seasons, including the Highland League Cup and several North of Scotland Cups. The local derby had been against Elgin City FC; however their promotion to the higher Scottish leagues in recent years has reduced the frequency of the matches. The town's junior football club is Lossiemouth United. RAF Lossiemouth also has a junior football club. In addition, the station has a rugby union and a cricket club that play in their respective North of Scotland leagues and a rugby league side that plays in the Scotland Rugby League as the Moray Eels; most station teams also play in their respective RAF competitions. The Moray Golf Club is the town's golf club and has two courses, the "Old Course" established in 1889, designed by Old Tom Morris who predicted that it would become the best in the north, and the 18 hole "New Course", designed by Sir Henry Cotton, opened in 1979.

Language
Lossiemouth, as with all of Moray and almost the entirety of Scotland, was previously Scottish Gaelic speaking. Following the 19th century, the process of Anglicization which had advanced for centuries saw it briefly adopt a dialect of the Scots language – closely related to the Doric dialect that was prevalent in nearby Aberdeenshire – which became dominant before being almost entirely displaced by standard Scottish English.

Many variations of the Doric dialect can be heard along the Grampian coast; identifiable dialects from towns as close as 5 mi apart (i.e. Lossiemouth and Hopeman) are not unheard of. Just as the Doric is in decline, however, so it is in Lossiemouth. The reasons for this include the demise of Lossiemouth as a fishing port where its fishermen used Scots extensively. In fishing towns such as Peterhead and Fraserburgh, Scots is still widely spoken. In Lossiemouth, though, the high level of employment at the RAF station and a large population of non-Scots (nearly 25%) has hastened this decline. Quite a lot of the words still remain in use but on the whole, Scottish English is increasingly spoken among the indigenous population.

Twin town

 * 🇩🇪 Hersbruck in Bavaria, Germany since 1972.

Notable Lossie-ites

 * James Ramsay MacDonald – first British Labour Prime Minister (1866–1937)
 * Malcolm MacDonald – Labour MP, Minister, diplomat and author (1901–1981)
 * Sergeant Alexander Edwards, VC – cooper and soldier (1885–1918)
 * Gordon Campbell, Baron Campbell of Croy, MC PC DL – soldier, diplomat, Conservative MP, Cabinet Minister and peer (1921–2005)
 * Stewart Imlach – professional footballer (1932–2001)
 * Alan Main – professional footballer and Scotland goalkeeper
 * David West, RSW – watercolour artist, gold prospector, sailor (1868–1936)
 * Meg Farquhar – first female professional golfer in Britain (1910–1988)
 * Jock Garden – Baptist minister, Australian politician and founder member of Australian Communist Party (1882–1968)
 * George Fraser – leading hybridiser of rhododendrons in British Columbia, Canada
 * Sir Alex Smith – former Head of Advanced Research, Rolls-Royce (1922 – 28 February 2003)
 * Peter Kerr – jazz musician, farmer, record producer and author
 * Very Rev Lewis Gordon (c.1750-c.1820) Moderator of the General Assembly of the Church of Scotland in 1815

Snippets
Two Royal Navy vessels, both named after the River Lossie, were involved in rescues following torpedo sinkings during the Second World War.
 * HMS Lossie, a River class frigate, was patrolling in the Indian Ocean. On 29 June 1944, the freighter Nellore was sunk and a week later HMS Lossie picked up 112 crewmen including the captain near the Chagos Archipelago and landed them at Addu Atoll.
 * HMS River Lossie (requisitioned Aberdeen drifter A332) picked up the master and 41 crew members of the merchant ship Cairnmona off Rattray Head on 30 October 1939 after she was sunk by a German U-boat while in a convoy bound from Montreal to Leith and Newcastle.

Lossiemouth was mentioned in Yes, Prime Minister; character Bernard Woolley thought it was a type of dog food.

title: Lossiemouth