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Pseudovector

In physics and mathematics, a pseudovector (or axial vector) is a quantity that behaves like a vector in many situations, but its direction does not conform when the object is rigidly transformed by rotation, translation, reflection, etc. This can also happen when the orientation of the space is changed. For example, the angular momentum is a pseudovector because it is often described as a vector, but by just changing the position of reference (and changing the position vector), angular momentum can reverse direction, which is not supposed to happen with true vectors (also known as polar vectors).

One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, that span the plane. The vector a × b is a normal to the plane (there are two normals, one on each side – the right-hand rule will determine which), and is a pseudovector. This has consequences in computer graphics, where it has to be considered when transforming surface normals. In three dimensions, the curl of a polar vector field at a point and the cross product of two polar vectors are pseudovectors.

A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics, in three dimensions, pseudovectors are equivalent to bivectors, from which the transformation rules of pseudovectors can be derived. More generally, in n-dimensional geometric algebra, pseudovectors are the elements of the algebra with dimension n − 1, written ⋀n−1Rn. The label "pseudo-" can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign-flip under improper rotations compared to a true scalar or tensor.

Physical examples
Physical examples of pseudovectors include torque, angular velocity, angular momentum, magnetic field, vorticity and magnetic dipole moment.

Consider the pseudovector angular momentum L = Σ(r × p). Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" (viewed as an ordinary vector) points to the right, but the actual angular momentum vector of the wheel (which is still turning forward in the reflection) still points to the left, corresponding to the extra sign flip in the reflection of a pseudovector.

The distinction between polar vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems. Consider an electric current loop in the z = 0 plane that inside the loop generates a magnetic field oriented in the z direction. This system is symmetric (invariant) under mirror reflections through this plane, with the magnetic field unchanged by the reflection. But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged.

In physics, pseudovectors are generally the result of taking the cross product of two polar vectors or the curl of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with (right-handed) pseudovectors and the right hand rule could be replaced by using (left-handed) pseudovectors and the left hand rule without issue. The (left) pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.

While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g. $$\mathbf{a}=(a_x,a_y,a_z)$$, and pseudovectors are represented in this form too. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by the exterior product of the two vectors, which yields a bivector which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.

Details
The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract vector space). Under the physics definition, a "vector" is required to have components that "transform" in a certain way under a proper rotation: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective of active transformations.) Mathematically, if everything in the universe undergoes a rotation described by a rotation matrix R, so that a displacement vector x is transformed to x = Rx, then any "vector" v must be similarly transformed to v = Rv. This important requirement is what distinguishes a vector (which might be composed of, for example, the x-, y-, and z-components of velocity) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box cannot be considered the three components of a vector, since rotating the box does not appropriately transform these three components.)

(In the language of differential geometry, this requirement is equivalent to defining a vector to be a tensor of contravariant rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the Einstein summation convention.)

A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the dyadic product, which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index. As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.

The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider improper rotations, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is inversion through a point in 3-dimensional space.) Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrix R, so that a position vector x is transformed to x = Rx. If the vector v is a polar vector, it will be transformed to v = Rv. If it is a pseudovector, it will be transformed to v = −Rv.

The transformation rules for polar vectors and pseudovectors can be compactly stated as

\begin{align} \mathbf{v}' & = R\mathbf{v} & & \text{(polar vector)} \\ \mathbf{v}' & = (\det R)(R\mathbf{v}) & & \text{(pseudovector)} \end{align} $$

where the symbols are as described above, and the rotation matrix R can be either proper or improper. The symbol det denotes determinant; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.

Behavior under addition, subtraction, scalar multiplication
Suppose v1 and v2 are known pseudovectors, and v3 is defined to be their sum, v3 = v1 + v2. If the universe is transformed by a rotation matrix R, then v3 is transformed to

\begin{align} \mathbf{v_3}' = \mathbf{v_1}'+\mathbf{v_2}' & = (\det R)(R\mathbf{v_1}) + (\det R)(R\mathbf{v_2}) \\ & = (\det R)(R(\mathbf{v_1}+\mathbf{v_2}))=(\det R)(R\mathbf{v_3}). \end{align} $$

So v3 is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any real number yields another pseudovector.

On the other hand, suppose v1 is known to be a polar vector, v2 is known to be a pseudovector, and v3 is defined to be their sum, v3 = v1 + v2. If the universe is transformed by an improper rotation matrix R, then v3 is transformed to

\mathbf{v_3}' = \mathbf{v_1}'+\mathbf{v_2}' = (R\mathbf{v_1}) + (\det R)(R\mathbf{v_2}) = R(\mathbf{v_1}+(\det R) \mathbf{v_2}). $$

Therefore, v3 is neither a polar vector nor a pseudovector (although it is still a vector, by the physics definition). For an improper rotation, v3 does not in general even keep the same magnitude:

 * $$|\mathbf{v_3}| = |\mathbf{v_1}+\mathbf{v_2}|, \text{ but } \left|\mathbf{v_3}'\right| = \left|\mathbf{v_1}'-\mathbf{v_2}'\right|$$.

If the magnitude of v3 were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in the weak interaction: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory. (See parity violation.)

Behavior under cross products
For a rotation matrix R, either proper or improper, the following mathematical equation is always true:
 * $$(R\mathbf{v_1})\times(R\mathbf{v_2}) = (\det R)(R(\mathbf{v_1}\times\mathbf{v_2}))$$,

where v1 and v2 are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation.)

Suppose v1 and v2 are known polar vectors, and v3 is defined to be their cross product, v3 = v1 × v2. If the universe is transformed by a rotation matrix R, then v3 is transformed to
 * $$\mathbf{v_3}' = \mathbf{v_1}' \times \mathbf{v_2}' = (R\mathbf{v_1}) \times (R\mathbf{v_2}) = (\det R)(R(\mathbf{v_1} \times \mathbf{v_2})) = (\det R)(R\mathbf{v_3}).$$

So v3 is a pseudovector. Similarly, one can show: This is isomorphic to addition modulo 2, where "polar" corresponds to 1 and "pseudo" to 0.
 * polar vector × polar vector = pseudovector
 * pseudovector × pseudovector = pseudovector
 * polar vector × pseudovector = polar vector
 * pseudovector × polar vector = polar vector

Examples
From the definition, it is clear that a displacement vector is a polar vector. The velocity vector is a displacement vector (a polar vector) divided by time (a scalar), so is also a polar vector. Likewise, the momentum vector is the velocity vector (a polar vector) times mass (a scalar), so is a polar vector. Angular momentum is the cross product of a displacement (a polar vector) and momentum (a polar vector), and is therefore a pseudovector. Torque is angular momentum (a pseudovector) divided by time (a scalar), so is also a pseudovector. Continuing this way, it is straightforward to classify any of the common vectors in physics as either a pseudovector or polar vector. (There are the parity-violating vectors in the theory of weak-interactions, which are neither polar vectors nor pseudovectors. However, these occur very rarely in physics.)

The right-hand rule
Above, pseudovectors have been discussed using active transformations. An alternate approach, more along the lines of passive transformations, is to keep the universe fixed, but switch "right-hand rule" with "left-hand rule" everywhere in math and physics, including in the definition of the cross product and the curl. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs. Nevertheless, there would be no physical consequences, apart from in the parity-violating phenomena such as certain radioactive decays.

Formalization
One way to formalize pseudovectors is as follows: if V is an n-dimensional vector space, then a pseudovector of V is an element of the (n − 1)-th exterior power of V: ⋀n−1(V). The pseudovectors of V form a vector space with the same dimension as V.

This definition is not equivalent to that requiring a sign flip under improper rotations, but it is general to all vector spaces. In particular, when n is even, such a pseudovector does not experience a sign flip, and when the characteristic of the underlying field of V is 2, a sign flip has no effect. Otherwise, the definitions are equivalent, though it should be borne in mind that without additional structure (specifically, either a volume form or an orientation), there is no natural identification of ⋀n−1(V) with V.

Another way to formalize them is by considering them as elements of a representation space for $$\text{O}(n)$$. Vectors transform in the fundamental representation of $$\text{O}(n)$$ with data given by $$(\mathbb{R}^n, \rho_{\text{fund}}, \text{O}(n))$$, so that for any matrix $$R$$ in $$\text{O}(n)$$, one has $$\rho_{\text{fund}}(R) = R$$. Pseudovectors transform in a pseudofundamental representation $$(\mathbb{R}^n, \rho_{\text{pseudo}}, \text{O}(n))$$, with $$\rho_{\text{pseudo}}(R) = \det(R)R$$. Another way to view this homomorphism for $$n$$ odd is that in this case $$\text{O}(n) \cong \text{SO}(n)\times \mathbb{Z}_2$$. Then $$\rho_{\text{pseudo}}$$ is a direct product of group homomorphisms; it is the direct product of the fundamental homomorphism on $$\text{SO}(n)$$ with the trivial homomorphism on $$\mathbb{Z}_2$$.

Geometric algebra
In geometric algebra the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.

The basic multiplication in the geometric algebra is the geometric product, denoted by simply juxtaposing two vectors as in ab. This product is expressed as:

 * $$ \mathbf {ab} = \mathbf {a \cdot b} +\mathbf {a \wedge b} \, $$

where the leading term is the customary vector dot product and the second term is called the wedge product or exterior product. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a multivector is a summation of k-fold wedge products of various k-values. A k-fold wedge product also is referred to as a k-blade.

In the present context the pseudovector is one of these combinations. This term is attached to a different multivector depending upon the dimensions of the space (that is, the number of linearly independent vectors in the space). In three dimensions, the most general 2-blade or bivector can be expressed as the wedge product of two vectors and is a pseudovector. In four dimensions, however, the pseudovectors are trivectors. In general, it is a (n − 1)-blade, where n is the dimension of the space and algebra. An n-dimensional space has n basis vectors and also n basis pseudovectors. Each basis pseudovector is formed from the outer (wedge) product of all but one of the n basis vectors. For instance, in four dimensions where the basis vectors are taken to be {e1, e2, e3, e4}, the pseudovectors can be written as: {e234, e134, e124, e123}.

Transformations in three dimensions
The transformation properties of the pseudovector in three dimensions has been compared to that of the vector cross product by Baylis. He says: "The terms axial vector and pseudovector are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual." To paraphrase Baylis: Given two polar vectors (that is, true vectors) a and b in three dimensions, the cross product composed from a and b is the vector normal to their plane given by c = a × b. Given a set of right-handed orthonormal basis vectors { eℓ }, the cross product is expressed in terms of its components as:

 * $$\mathbf {a} \times \mathbf{b} = \left(a^2b^3 - a^3b^2\right) \mathbf {e}_1 + \left(a^3b^1 - a^1b^3\right) \mathbf {e}_2 + \left(a^1b^2 - a^2b^1\right) \mathbf {e}_3 ,$$

where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by a ∧ b. In this context of geometric algebra, this bivector is called a pseudovector, and is the Hodge dual of the cross product. The dual of e1 is introduced as e23 ≡ e2 ∧ e3, and so forth. That is, the dual of e1 is the subspace perpendicular to e1, namely the subspace spanned by e2 and e3. With this understanding,

 * $$ \mathbf{a} \wedge \mathbf{b} = \left(a^2b^3 - a^3b^2\right) \mathbf {e}_{23} + \left(a^3b^1 - a^1b^3\right) \mathbf {e}_{31} + \left(a^1b^2 - a^2b^1\right) \mathbf {e}_{12} \ . $$

For details, see . The cross product and wedge product are related by:

 * $$\mathbf {a} \ \wedge \ \mathbf{b} = \mathit i \ \mathbf {a} \ \times \ \mathbf{b} \ ,$$

where is called the unit pseudoscalar. It has the property:

 * $$\mathit{i}^2 = -1 \ . $$

Using the above relations, it is seen that if the vectors a and b are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the cross product are invariant. On the other hand, if the components are fixed and the basis vectors eℓ are inverted, then the pseudovector is invariant, but the cross product changes sign. This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.

Note on usage
As an aside, it may be noted that not all authors in the field of geometric algebra use the term pseudovector, and some authors follow the terminology that does not distinguish between the pseudovector and the cross product. However, because the cross product does not generalize to other than three dimensions, the notion of pseudovector based upon the cross product also cannot be extended to a space of any other number of dimensions. The pseudovector as a (n – 1)-blade in an n-dimensional space is not restricted in this way.

Another important note is that pseudovectors, despite their name, are "vectors" in the sense of being elements of a vector space. The idea that "a pseudovector is different from a vector" is only true with a different and more specific definition of the term "vector" as discussed above.

title: Pseudovector


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River Mersey

The River Mersey is a major river in North West England. Its name derives from Old English and means "boundary river", possibly referring to its having been a border between the ancient kingdoms of Mercia and Northumbria. For centuries it has formed part of the boundary between the historic counties of Lancashire and Cheshire.

The Mersey starts at the confluence of the River Tame and River Goyt in Stockport. It flows westwards through south Manchester, then into the Manchester Ship Canal at Irlam, becoming a part of the canal and maintaining its water levels. After 4 mi it exits the canal, flowing towards Warrington where it widens. It then narrows as it passes between Runcorn and Widnes. The river widens into a large estuary, which is 3 mi across at its widest point near Ellesmere Port. The course of the river then turns northwards as the estuary narrows between Liverpool and Birkenhead on the Wirral Peninsula to the west, and empties into Liverpool Bay. In total the river flows 69 mi.

Part of the Mersey Railway, a rail tunnel between Birkenhead and Liverpool opened in 1886. Two road tunnels pass under the estuary from Liverpool: the Queensway Tunnel opened in 1934 connecting the city to Birkenhead, and the Kingsway Tunnel, opened in 1971, to Wallasey. A road bridge, completed in 1961 and later named the Silver Jubilee Bridge, crosses between Runcorn and Widnes, adjacent to the Runcorn Railway Bridge which opened in 1868. A second road bridge, the Mersey Gateway, opened in October 2017, carrying a six-lane road connecting Runcorn's Central Expressway with Speke Road and Queensway in Widnes. The Mersey Ferry operates between Pier Head in Liverpool and Woodside in Birkenhead and Seacombe, and has become a tourist attraction offering cruises that provide an overview of the river and surrounding areas.

Water quality in the Mersey was severely affected by industrialisation, and in 1985, the Mersey Basin Campaign was established to improve water quality and encourage waterside regeneration. In 2009 it was announced that the river is "cleaner than at any time since the industrial revolution" and is "now considered one of the cleanest in the UK". The Mersey Valley Countryside Warden Service manages local nature reserves such as Chorlton Ees and Sale Water Park.

The river gave its name to Merseybeat, developed by bands from Liverpool, notably the Beatles. In 1965 it was the subject of the top-ten hit single "Ferry Cross the Mersey" by Gerry and the Pacemakers.

Etymology
Its name is derived from the Anglo-Saxon mǣres, 'of a boundary' and ēa, 'a river'. The Mersey was possibly the 'border river' between Mercia and Northumbria. Its Welsh name is Afon Merswy.

Course
The Mersey is formed from three tributaries: the River Etherow, the River Goyt and the River Tame. The modern accepted start of the Mersey is at the confluence of the Tame and Goyt, in central Stockport, Greater Manchester. However, older definitions, and many older maps, place its start a few miles up the Goyt at Compstall; for example the 1911 Encyclopædia Britannica states "It is formed by the junction of the Goyt and the Etherow a short distance below Marple in Cheshire on the first-named stream." The 1784 John Stockdale map shows the River Mersey extending to Mottram, and forming the boundary between Cheshire and Derbyshire. In the west of Stockport it flows at the base of a cliff below the road called Brinksway before reaching flat country. An early Ordnance Survey map indicates the spring at Red Hole on the border of Cheshire and Yorkshire at the head of one of the Etherow's longest headwaters, as being the Source of the River Mersey.

Stockport to Warrington

From Central Stockport the river flows through or past Heaton Mersey, Didsbury, Northenden, Chorlton-cum-Hardy, Stretford, Sale, Ashton on Mersey, Urmston and Flixton, then at Irlam flows into the Manchester Ship Canal, which is the canalised section of the River Irwell at this point. The old course of the Mersey has been obliterated by the canal past Hollins Green to Rixton although the old river bed can be seen outside Irlam and at Warburton. At Rixton the River Bollin enters the canal from the south and the Mersey leaves the canal to the north, meandering through Woolston, where the ship canal company's dredgings have formed the Woolston Eyes nature reserve, and on to Warrington.

During the construction of the Manchester Ship Canal, the original course of the river through Wilderspool (which is north of Stockton Heath and Lower Walton) was diverted westward into a canalised section. Wilderspool Causeway remains the only remnant of the original eastern bank of the Mersey at this point in Warrington. Part of the original river channel became Warrington Dock (Walton Lock) in the ship canal. In the 1960s, the former river channel was filled. The area is now Wilderspool Causeway Park. At the western end of the canalised section in Warrington is the old entrance to the former Runcorn to Latchford Canal. This waterway was abandoned with the construction of the ship canal. It was constructed because the Mersey is tidal from Howley Weir in Warrington. The canal let river traffic reach Warrington during low tides.

During high spring tides, water levels often top the Howley Weir. Before construction of the ship canal, work to improve navigation on the Mersey included Woolston New Cut, bypassing a meander, and Howley Lock for craft to avoid the weir; the new cut and lock are still evident. The island formed between the weir and the lock is known locally as "Monkey Island".

Runcorn Gap
West of Warrington the river widens, and then narrows as it passes through the Runcorn Gap between the towns of Runcorn and Widnes, in Halton. The Manchester Ship Canal passes through the gap to the south of the river. The gap is bridged by the Silver Jubilee Bridge and Runcorn Railway Bridge. Another crossing, the Mersey Gateway road bridge opened in October 2017, and is situated about 1 mile upstream from the Runcorn Gap where the river is considerably wider.

Estuary
From the Runcorn Gap, the river widens into a large estuary, which is 3 mi wide at its widest point near Ellesmere Port. The course of the river then heads north, with Liverpool to the east and the Wirral Peninsula to the west. The Manchester Ship Canal enters the river at Eastham Locks. The eastern part of the estuary is much affected by silting, and part of it is marked on modern maps as dry land rather than tidal. The wetlands are of importance to wildlife, and are listed as a Ramsar site.

Most of the conurbation on both sides of the estuary is known as Merseyside. The estuary narrows between Liverpool and Birkenhead, where it is constricted to a width of 0.7 mi, between Albert Dock in Liverpool and the Woodside ferry terminal in Birkenhead. On the Liverpool side, Liverpool Docks stretch for over 7.5 mi, the largest enclosed interconnected dock system in the world. American author Herman Melville described Liverpool Docks as being comparable to the pyramids in the vast scale of their construction. The docks were built out into the river bed. The Mersey Docks and Harbour Board used granite from a quarry it owned in Scotland for construction of the quays. Birkenhead grew quickly in the 19th century following the introduction of steamships, the earliest being the wooden paddle steamer Elizabeth in 1815. Docks were developed along with a shipbuilding industry, flour milling and soap manufacture on the river's Cheshire bank.

Seaforth Dock, a freeport on the Liverpool side of the estuary where it meets Liverpool Bay, opened in 1971. The dock deals with around 500,000 containers, over 1.2 million tonnes of oil, over 2.5 million tonnes of grain and animal feed, 452,000 tonnes of wood per year. About 25% of all container traffic between the UK and USA passes through the port making it one of the most successful in the world and known as the "Atlantic Gateway". Liverpool was the first UK port with radar assisted operations.

The river empties into Liverpool Bay on the Irish Sea, after a total course of 69 mi. From 4 m neap tide to 10 m spring tide, the River Mersey has the second highest tidal range in Britain – second only to the River Severn. The narrows in the river estuary between Dingle Point on the Liverpool banks to New Brighton on the Wirral, forces water to flow faster creating a deep channel along the section of river.

For 200 years an admiral was appointed as a conservator to ensure the river remained navigable. Mary Kendrick was the first woman to the post and she had spent years studying a Mersey barrage in the 1980s.

Taylor's Bank is a large sandbank extending out to sea on the north side of the channel entrance to the river on which many ships have come to grief over the years.

River crossings
Historically the lowest bridging point on the Mersey was at Warrington where there has been a bridge since medieval times. The first ferry across the estuary was introduced in medieval times by monks from Birkenhead Priory. They transported travellers or accommodated them at the priory in bad weather.

Countless people died on the River Mersey until the advent of safe passage via bridges and tunnels. Many of these deaths were of people who earned their living on the river, either as mariners or dock workers. The majority of mariners to die worked on the Mersey flat boats and drowned due to the weather conditions or poor craft maintenance, although many ordinary civilians perished too. Another ferry existed at Runcorn Gap and by today's safety standards was highly dangerous. Passengers had to traverse wooden planks over the mud flats to reach the ferry boats which themselves were often poorly maintained and leaking. Even the steps down to the river bank were described as "beautifully slippery". In the dark the ferry was particularly worrying as there were few or no lights and the journey was only undertaken on a "needs must" basis. Complaints about the appalling conditions were reported in the local and national press consistently for over 40 years.

In the early 19th century steam operated ferries were introduced. The Mersey Ferry, managed and operated by Merseytravel, operates between Pier Head in Liverpool and Woodside in Birkenhead and Seacombe. It has become a tourist attraction offering cruises that provide an overview of the river and surrounding areas.

The Mersey Railway completed its tunnel through the estuary's underlying Triassic sandstone using manual labour in 1885. Intended as a pneumatic railway, the company opted for steam trains from its opening until it was electrified in 1903. The centre of the running tunnel is between 30 ft and 70 ft below the river bed. The railway is now part of the Merseyrail network. Two road tunnels pass under the estuary from Liverpool. By road, the Queensway Tunnel opened in 1934 connecting the city to Birkenhead, and the Kingsway Tunnel, opened in 1971, connects with Wallasey.

Further upstream, the Runcorn Railway Bridge over the river at Runcorn Gap was built in the 1860s for the London and North Western Railway on the mainline between London and Liverpool. It had a cantilevered footway providing an alternative crossing to a ferry. In 1905 the now demolished Widnes-Runcorn Transporter Bridge opened and took cars and passengers via a cable car. The Silver Jubilee Bridge, completed in 1961, is immediately adjacent.

East of Warrington, the M6 motorway crosses the river and the Manchester Ship Canal on the Thelwall Viaduct. When the viaduct opened in 1963, it was the longest motorway bridge in England. A second viaduct alongside opened in 1995; northbound traffic uses the old viaduct and southbound the new.

A new 6-lane bridge, the "Mersey Gateway", between Runcorn and Widnes began its construction phase in May 2014 and opened just after midnight on 14 October 2017. At the same time the Silver Jubilee Bridge was closed to undergo maintenance for approximately 6–12 months, but will remain open to pedestrians and cyclists. The new bridge uses tolls and the existing Silver Jubilee Bridge will also be tolled.

Environment
Water quality in the Mersey was severely affected by industrialisation, and in 1985, the Mersey Basin Campaign was established to improve water quality and encourage waterside regeneration. In 2002, oxygen levels that could support fish along the entire length were recorded for the first time since industry began on the Mersey.

In 2009 it was announced that the river is "cleaner than at any time since the industrial revolution" and is "now considered one of the cleanest [rivers] in the UK".

Sediment pollution
Persistent organic and mercury (Hg) pollution contained within sediments of the Mersey estuary have been evaluated by British Geological Survey. Mersey river sediments from outer to inner estuary (Alfred Dock to Widnes) contain a variety of common organic pollutants, including polyaromatic hydrocarbons (PAH) with concentrations which fall in the middle of the range of similarly industrial-urban river-estuaries. The distribution of individual PAH compounds suggests that the Mersey has contaminants mainly derived from combustion sources such as vehicle exhaust as well as coal burning. The distribution of the toxic heavy metal Mercury (Hg) has been assessed by measuring 203 sediments taken from shallow cores (0.4 – 1.6 m) extracted from both the main river and adjacent salt marshes. The average amount of Hg in the Mersey was found to be 2 mg/kg with the highest amounts of 5 mg/kg occurring below the surface at concentrations harmful to sediment dwelling biota. The vertical rise and fall in Hg pollution observed at four Mersey salt marshes indicated a decline in metal pollution since the 1980s.

Fauna
Salmon have returned to the river and have been seen jumping at Woolston and Howley Weirs between September and November. Salmon parr and smolt have been caught in the Mersey's tributaries, the River Goyt and the River Bollin. Atlantic grey seals from Liverpool Bay occasionally venture into the estuary along with bottlenose dolphin and harbour porpoise. Otter tracks have been observed near Fiddlers Ferry. Other fish in the estuary include cod, whiting, dab, plaice and flounder. Spiny dogfish, mackerel and tope feed on shrimp and whitebait in the estuary. For ducks, the estuary is important to common shelducks and Eurasian teal. Waders include redshank, black-tailed godwit, dunlin, pintail and turnstone.

Environmental designations
The section of the estuary between Runcorn Railway Bridge and Bromborough, including Hale Duck Decoy and Mount Manisty, is designated as a Site of Special Scientific Interest, the SSSI citation describing it as an "internationally important site for wildfowl". There are further SSSIs at New Ferry and Mersey Narrows. The Mersey Estuary, and the Mersey Narrows and North Wirral Foreshore, are also Ramsar sites.

Navigation
Capt. William Gill of the Isle of Man Steam Packet Company, charted a safe, navigable channel (the Victoria Channel) through the treacherous uncharted waters of the estuary in 1836.

Since the construction of the Manchester Ship Canal, large commercial vessels do not usually navigate the estuary beyond Garston on the north bank, or the locks into the ship canal at Eastham. Deep-water channels are maintained to both. Until the early 20th century, commercial traffic bound for further upstream carried cargo in large flat-bottomed sailing barges known as Mersey Flats to Howley Wharf in Warrington and (via the Sankey Canal) to St Helens. Motor barges delivered to riverside factories at Warrington until at least the 1970s, but nowadays only pleasure craft and yachts use the upper estuary and the tidal river where a number of sailing clubs are based. On most high tides, seagoing yachts with masts raised can navigate as far upstream as Fiddlers Ferry – about 3.1 mi downstream of Warrington – where there is a small marina accessed via a river lock. Although river craft can continue upstream to Howley Weir, there are no landing or mooring facilities. Before construction of the ship canal, a lock bypassing Howley Weir allowed navigation further upstream via a straight "cut" avoiding a meander around Woolston.

Recreation
The Mersey Valley Countryside Warden Service manages local nature reserves such as Chorlton Ees and Sale Water Park recreational sites and provides an educational service along the Mersey from Manchester to the Manchester Ship Canal.

It is possible to canoe on parts of the river between Stockport and Carrington. Liverpool Sailing Club located at Garston Coastal Park on the north bank of the estuary has a 1000 feet slipway giving access to river for water sports.

The wooded suburban stretch of the river from above Howley Weir to Woolston is also used for recreational and competitive rowing, operated from the Warrington Rowing Club.

Angling has become popular on some stretches of the river as fish such as perch, barbel, grayling, carp, roach, chub, trout, pike, bream and dace have been caught. Warrington Anglers Association have fishing rights on a large stretch of the River through most of Warrington. Prince Albert Angling Society also have a small stretch near Rixton.

The river has also faced problems regarding the poaching of fish despite a strict 'catch and release' policy imposed on anglers for ecological reasons which applies to most UK waterways.

Mersey barrage
Proposals continue for the construction of the Mersey Barrage, a tidal scheme to generate electricity and create another crossing of the river. Very high spring tides can generate a tidal bore from Hale as far upstream as Warrington. On 7 December 2022, the Liverpool City Region mayor announced cooperation between the City Region and K-water of South Korea, who built and operates the Sihwa Lake Tidal Power Station, in order to construct a similar operation on the River Mersey.

In popular culture
The river gave its name to Merseybeat, developed by bands from Liverpool, notably the Beatles. In 1965 it was the subject of the top-ten hit single "Ferry Cross the Mersey" by Gerry and the Pacemakers, and a musical film of the same name. The Liverpool poets published an anthology of their work, The Mersey Sound, in 1967.

The river's stretch through Manchester is the main theme in the Stone Roses song "Mersey Paradise", included on the B-side of the 12" version of their 1989 hit "She Bangs the Drums", and on their 1992 compilation album Turns into Stone.

The Tall ships' fleet has visited the Mersey on four occasions, first in 1984, then in 1992, 2008 and 2012.

The traditional song "Leaving of Liverpool" mentions the river in its opening line.

Religious significance
The Mersey is considered sacred by British Hindus, and worshipped in a similar way to the River Ganges. Festival of Immersion ceremonies are held annually on the river, in which clay figures representing the Hindu Lord Ganesha, the elephant deity riding a mouse, are submerged in the river from a ferry boat. Followers throw flowers, pictures and coins into the river.

Tributaries
From its lowest point, moving upstream, confluences and tributaries of the Mersey catchment include:

 * The Birket
 * River Fender
 * Prenton Brook
 * Arrowe Brook
 * Greasby Brook
 * Newton Brook
 * River Dibbin
 * Clatter Brook
 * River Gowy
 * Hornsmill Brook
 * River Weaver
 * River Dane
 * River Wheelock
 * River Croco
 * Ram's Brook
 * Bowers Brook
 * Sankey Brook
 * Padgate Brook
 * Morris Brook
 * Spittle Brook
 * Thelwall Brook
 * Fishington Brook
 * River Bollin
 * River Dean
 * Marsh Brook
 * Red Brook
 * Glaze Brook/River Glaze
 * River Irwell
 * River Medlock
 * River Roch
 * River Beal
 * River Spodden
 * River Irk
 * River Croal
 * Old Eea Brook
 * Stromford Brook
 * Chorlton Brook
 * Barrow Brook
 * Gatley Brook
 * Micker Brook
 * Tin Brook
 * River Tame
 * River Goyt
 * River Etherow
 * River Sett
 * River Kinder

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

title: River Mersey


text:
La Tosca

La Tosca is a five-act drama by the 19th-century French playwright Victorien Sardou. It was first performed on 24 November 1887 at the Théâtre de la Porte Saint-Martin in Paris, with Sarah Bernhardt in the title role. Despite negative reviews from the Paris critics at the opening night, it became one of Sardou's most successful plays and was toured by Bernhardt throughout the world in the years following its premiere. The play itself had dropped from the standard theatrical repertoire by the mid-1920s, but its operatic adaptation, Giacomo Puccini's Tosca, has achieved enduring popularity. There have been several other adaptations of the play including two for the Japanese theatre and an English burlesque, Tra-La-La Tosca (all of which premiered in the 1890s) as well as several film versions.

La Tosca is set in Rome on 17 June 1800 following the French victory in the Battle of Marengo. The action takes place over an eighteen-hour period, ending at dawn on 18 June 1800. Its melodramatic plot centers on Floria Tosca, a celebrated opera singer; her lover, Mario Cavaradossi, an artist and Napoleon sympathiser; and Baron Scarpia, Rome's ruthless Regent of Police. By the end of the play, all three are dead. Scarpia arrests Cavaradossi and sentences him to death in the Castel Sant'Angelo. He then offers to spare her lover if Tosca will yield to his sexual advances. She appears to acquiesce, but as soon as Scarpia gives the order for the firing squad to use blanks, she stabs him to death. On discovering that Cavaradossi's execution had in fact been a real one, Tosca commits suicide by throwing herself from the castle's parapets.

Background and premiere
Victorien Sardou's grandfather had served as a surgeon with Napoleon's army in Italy, and Sardou retained a lifelong interest in the French Revolution and the French Revolutionary Wars. In addition to La Tosca, six of his other plays were set against the events of those times: Monsieur Garat (1860), Les Merveilleuses (1873), Thermidor (1891), Madame Sans-Gêne (1893), Robespierre (1899), and Pamela (1898). He was known for the historical research which he used to inform his plays and had a private research library of over 80,000 books including Piranesi's etchings of late 18th century Rome, where La Tosca is set.

Sardou wrote La Tosca specifically for Sarah Bernhardt. She was in her mid-40s by then and France's leading actress. In 1883, she had also taken over the lease on the Théâtre de la Porte Saint-Martin, where La Tosca was to premiere. It was the third play Sardou had written specifically for her. Both their first collaboration, Fédora (1882), and their second, Théodora (1884), had been highly successful. Pierre Berton, who played Baron Scarpia, had been Bernhardt's on and off lover for many years and a frequent stage partner. The elaborate sets for the production were made by a team of designers and painters who had worked with Sardou before: Auguste Alfred Rubé, Philippe Chaperon, Marcel Jambon, Enrico Robecchi, Alfred Lemeunier, and Amable Petit. The costumes were designed by Théophile Thomas, who also designed Sarah Bernhardt's costumes for Hugo's Ruy Blas, Sardou's Cléopâtre and Théodora, and Barbier's Jeanne d'Arc.

The period leading up to the premiere was not without problems. As had happened before, once word got out of a new Sardou play, another author would accuse him of plagiarism. In the 1882 caricature of Sardou (left), one of the signs on the wall states, "Idées des autres" ("Ideas of others") and another, "Bien d'auteur" ("Author's rights"). This time Ernest Daudet (a brother of Alphonse Daudet) made the accusation, claiming that four years earlier, he and Gilbert-Augustin Thierry had written a play, Saint Aubin, which takes place in Paris on the day after the Battle of Marengo (roughly the same time-setting as La Tosca) and whose heroine (like Tosca) is a celebrated opera singer. He also claimed that he had read the play to Sarah Bernhardt and Félix Duquesnel, the director of the Théâtre de la Porte Saint-Martin. Nevertheless, he said he would "graciously permit" Sardou's play to go ahead, and had brought up the issue solely to avoid being accused of plagiarism should Saint-Auban ever be produced. Sardou, in turn, issued a robust denial in the French papers. As the play neared its premiere, Bernhardt discovered to her fury that Sardou had sold the rights for the first American production of the play to the actress Fanny Davenport and threatened to walk out. Bernhardt was eventually pacified and rehearsals continued.

The Théâtre de la Porte Saint-Martin was packed for the opening night on 24 November 1887, although many in the audience already knew the ending before the curtain went up. While journalists were usually invited to dress-rehearsals, they were expected not to publish details of the play before the premiere. However, the Parisian journal, Gil Blas, had published a complete description of the plot on the morning of 24 November. (Following the premiere, Sardou brought a successful suit for damages against the paper. ) At the end of the performance, Pierre Berton (Scarpia) came on stage for the customary presentation of the author to the audience. As he began his introduction, a large part of the audience interrupted him shouting, "Bernhardt, Bernhardt!" After three failed attempts, he went backstage and asked Bernhardt to come out. She refused to do so until Sardou had been introduced. Berton finally succeeded, after which Bernhardt appeared to thunderous applause and cries of "Vive Sarah!"

Characters
Three minor characters in La Tosca are real historical figures: Queen Maria Carolina; Prince Diego Naselli, the Governor of Rome; and the composer, Giovanni Paisiello. However, their treatment in the play is not always historically accurate. On the day the play takes place, Queen Maria Carolina was actually on her way to Austria and staying in Livorno, not Rome. Paisiello was a Neapolitan court composer, but at the time of the play he was under suspicion for anti-Royalist sympathies, making him a highly unlikely candidate for Maria Carolina's gathering in Act 2. According to Deborah Burton, another minor character, Princesse Orlonia, is probably based on Princess Torlonia. Although their names and backgrounds contain historical allusions, the four main protagonists, Cesare Angelotti, Mario Cavaradossi, Floria Tosca, and Baron Scarpia are fictional. Their backgrounds are revealed in the conversations between Angelotti and Cavaradossi in Acts 1 and 3. Cesare Angelotti had been a wealthy landowner in Naples and defender of the short-lived Neapolitan Republic. When it fell to the British forces and Ferdinand IV was returned as ruler, he fled to Rome where he became one of the Consuls of the equally short lived Roman Republic. He is a wanted man, not only for his revolutionary activities but also for a youthful dalliance in London, where he had an eight-day liaison with Emma Hamilton. In those days she had been a prostitute going by the name of Emma Lyon, but by the time of the play she had become the wife of the British Envoy to Naples, William Hamilton, and was a favourite of Queen Maria Carolina. Determined to avoid a scandal, the Queen demanded that he be returned to Naples and hanged. He was languishing in Rome's Castel Sant'Angelo, when his sister Giulia, the Marquise Attavanti, helped him to escape. According to historian Susan Vandiver Nicassio, Angelotti was partly based on Liborio Angelucci, who had briefly been a Consul of the Roman Republic, although the resemblance in terms of their life histories ends there. Another influence on the choice of surname may have been Nicola Antonio Angeletti (1791–1870), a prominent Italian revolutionary and member of the Carbonari.

Mario Cavaradossi is descended from an old Roman family, but was born in France, where his father had lived most of his life. The family still had a palazzo on the Piazza di Spagna in Rome and once owned the country villa which Cavaradossi now rents. His father had strong ties with Diderot and d'Alembert, and his mother was a grand-niece of the French philosopher Helvétius. Cavaradossi studied art in Paris with Jacques-Louis David and lived in David's atelier during the French Revolution. When he visited Rome in 1800 to settle his father's estate, he met and fell in love with the celebrated opera singer Floria Tosca, and decided to prolong his stay. He soon gained a reputation as a free-thinker and Bonapartist. Even his mustache was suspect. Tosca's confessor told her it marked him as a revolutionary. To deflect these suspicions, he offered to do a painting in the church of Sant'Andrea al Quirinale for free. Nicassio has speculated that one of the influences on Sardou's choice of name was the extremely similar name Caravadossi, a noble Italian family from Nice, the birthplace of Garibaldi, and at several points in its history under Italian control. One of the Caravadossi descendants fought in the 19th century Italian Wars of Independence.

Floria Tosca is an orphan from Verona, where she had been found as a child, roaming the hillsides and herding sheep. The Benedictine monks took her in and educated her. The convent organist gave her singing lessons, and by the time she was sixteen, her church performances had made her a local celebrity. The composer Domenico Cimarosa went to hear her and wanted her to go on stage. The monks opposed this, but after she was presented to the Pope, he too declared that she should become an opera singer. Four years later she made her debut in the title role of Paisiello's Nina and went on to sing at La Scala, La Fenice, and the Teatro San Carlo to great acclaim. When Cavaradossi met her she was singing at the Teatro Argentina in Rome. As soon as her engagement at the theatre was over, she and Cavaradossi planned to leave for Venice, where she had a contract to sing at La Fenice. Sardou took a long time to decide on her name and may have finally been influenced by Saint Tosca, who is particularly revered in Verona. The 8th-century church dedicated to her there is one of the oldest in the Veneto region.

Baron Vitellio Scarpia is from Sicily, where he was known for his ruthless law enforcement. When Naples took control of Rome in 1799, he was appointed the city's Regent of Police, and quickly gained a reputation for the cruelty and licentiousness that lay beneath his seemingly courteous exterior. Angelotti characterises him as a religious hypocrite and an "impure satyr" from whom no woman is safe. Before Scarpia set his sights on Floria Tosca, he had tried to force himself on Angelotti's sister, who fled from him in terror. According to Nicassio, Sardou may have chosen his name for its similarity to "Sciarpa", the nickname of Gherardo Curci, a bandit who led irregular troops fighting on behalf of the monarchy in Naples and was made a baron by Ferdinand IV in 1800.

Historical context
La Tosca is set against the background of the French Revolutionary Wars, the establishment of the Roman Republic, and its subsequent fall in 1799 when the French withdrew from Rome. Following the French withdrawal, Rome was controlled by the Kingdom of Naples, supported by the British and Austrians. However, the fighting continued elsewhere in Italy. The French troops had been defeated by the Austrians at the Siege of Genoa on 4 June 1800. Then on 14 June 1800, three days before the play begins, Napoleon's troops fought the Austrian forces at the Battle of Marengo. Although out-numbered, the French were ultimately victorious, despite early reports to the contrary. News of the surprise victory reached Rome on 17 June, the time setting for the play.

Act 1
The church of Sant'Andrea al Quirinale in Rome on the afternoon of 17 June 1800

Gennarino (Cavaradossi's manservant) and Eusèbe (the sacristan) discuss Cavaradossi's relationship with Tosca, his Republican and Bonapartist sympathies, and the apparent defeat of the French army at Marengo. Cavaradossi arrives to work on his painting of Mary Magdalen. When Gennarino and Eusèbe leave, Angelotti, a Republican fugitive who has escaped from the Castel Sant'Angelo emerges from his hiding place in his family's chapel. His sister, the Marquise Attavanti, had visited the day before to leave him supplies and women's clothes to disguise himself, including a fan to hide his face. Cavaradossi recalls seeing a beautiful blond woman in the church the previous day and tells how she inspired his painting. Tosca arrives and Angelotti quickly returns to his hiding place. Tosca, who is dark-haired, becomes jealous when she sees Cavaradossi's painting of a blonde woman, but he reassures her of his love. After she departs, Cavaradossi and Angelotti quickly leave for Cavaradossi's country villa. Baron Scarpia and his police enter the church searching for Angelotti. Scarpia finds the fan left by the Marquise Attavanti and keeps it. Worshippers arrive for the Te Deum which has been ordered to give thanks for the French defeat.

Act 2
A large chamber in the Farnese Palace on the evening of 17 June 1800

At the gambling tables, Vicomte de Trévilhac, Capréola, Trevulce and the Marquis Attavanti (all supporters of the Kingdom of Naples), discuss the French defeat at Genoa earlier that month, their apparent defeat at Marengo, and the disappearance of Angelotti and Cavaradossi. Princesse Orlonia and other ladies of the court join them. All discuss the cantata by Paisiello which Tosca will sing later that evening as part of the victory celebrations. Baron Scarpia arrives and there is further discussion of Angelotti's escape, cut short by the arrival of Tosca. Queen Marie Caroline enters for the performance of the cantata accompanied by Paisiello, Prince Diego Naselli, courtiers, musicians, Austrian army officers, and monsignors. She reiterates her demand that Scarpia capture Angelotti and have him hanged. Scarpia must now find the fugitive's hiding place as quickly as possible. Hoping to provoke Tosca into leading him to Cavaradossi and Angelotti, he takes her aside and shows her the Marquise Attavanti's fan, intimating that she and Cavaradossi are lovers. Tosca is overcome with jealousy. As the cantata performance is about to begin, couriers arrive with a letter announcing that the French had been victorious at the Battle of Marengo after all. The Queen faints. Tosca throws the pages of her score into the air and rushes out with her maid. Scarpia orders his men to follow her carriage.

Act 3
Cavaradossi's country villa on the night of 17 June 1800

Cavaradossi tells Angelotti of a chamber in an ancient Roman well on the property where he can hide until he makes his escape. It had been used by one of Cavaradossi's ancestors when he fled Rome after stabbing a Medici. Tosca arrives to confront her lover about the fan Scarpia had shown her. Cavaradossi and Angelotti explain everything and she realizes with horror that she has been duped into leading Scarpia to them. On hearing the arrival of Scarpia and his men, Angelotti seeks refuge in the well. Scarpia demands to know where Angelotti is hidden. When Tosca and Cavardossi refuse to tell, Cavaradossi is taken off to be interrogated by the Procureur and tortured by Scarpia's assistant if he refuses to answer. Scarpia describes the torture device in great detail to Tosca, who is then made to listen to her lover's screams. Unable to bear it any longer, she reveals the hiding place, much to Cavaradossi's fury. Rather than be captured, Angelotti takes poison concealed in his ring. Scarpia orders his men to take Cavaradossi to the Castel Sant'Angelo for execution and orders Tosca to be brought there as well.

Act 4
Scarpia's apartments in the Castel Sant'Angelo in the hours of darkness before the dawn of 18 June 1800

Scarpia is eating supper in a room lit only by two candles and a candelabrum on his table. There is a prayer stool and a crucifix in an alcove near his bed. He orders Tosca, who has been locked in another room of the castle, to be brought to him. When she arrives, he tells her that Cavaradossi is to be hanged at dawn. He also tells her of his intense attraction to her and offers to spare Cavaradossi if she agrees to sleep with him. Tosca calls him a wild animal and repels his advances in disgust, which only serves to increase his desire. Scarpia then takes her to the window and shows her the scaffold awaiting her lover. Tosca finally says that she will agree to his terms, but only after she has proof that Cavaradossi will be spared. Scarpia calls in Spoletta and in front of Tosca instructs him to stage a mock execution by firing squad with blanks in the riflemen's guns. After Spoletta leaves, Tosca demands that Scarpia also give her a document granting safe conduct out of the Roman States. As soon as he signs the document and starts to kiss her, she grabs a knife from the supper table and stabs Scarpia to death. Tosca removes the safe conduct from his hand and starts to leave, but then turns back. She places the two lighted candles on each side of Scarpia's body and puts the crucifix on his chest before quietly slipping out of the room.

Act 5
The chapel at the Castel Sant'Angelo and a platform on the roof of the castle at dawn on 18 June 1800

Spoletta and his men awaken Cavaradossi in the chapel where he is being held to tell him that he has a visitor. Tosca arrives and rushes into her lover's arms. She begs his forgiveness for having revealed Angelotti's hiding place, and he in turn asks forgiveness for his anger at the time. She explains that the execution will only be a mock one and they will be able to escape from Rome. Spoletta confirms this and leaves to prepare the firing squad. Alone with Cavaradossi, Tosca tells him that she has killed Scarpia. Spoletta returns to take Cavaradossi to the platform where the firing squad awaits and tells Tosca to remain behind. After a few minutes, Tosca goes out onto the platform and sees Cavaradossi lying on the ground. She turns him over and discovers that he is dead. The bullets were real. Spoletta reveals that he was in fact following Scarpia's orders which contained the coded message to shoot him "like we shot Count Palmieri". Distraught at Scarpia's betrayal, Tosca screams "And I cannot even kill him again!" At first Spoletta and Schiarrone think she has gone mad, but an officer arrives and confirms that Scarpia has been murdered. As Spoletta lunges towards her, Tosca climbs onto the castle parapets and throws herself off.

Performance history
La Tosca had an opening run in Paris of 200 performances. Sarah Bernhardt, along with the original Cavaradossi (Camille Dumény) and Baron Scarpia (Pierre Berton), then starred in the London premiere in July 1888 at the Lyceum Theatre. She would continue to be closely associated with the play until well into the 20th century, touring it around the world from 1889, including performances in Egypt, Turkey, Australia and several countries in Latin America. It was during her 1905 tour to Rio de Janeiro that she injured her leg jumping from the parapets in the final scene. The wound never healed properly and ultimately led to amputation of her leg ten years later. Bernhardt gave the first American performance of La Tosca in the original French at New York's Garden Theater on 5 February 1891 and took the play to many other American cities, aways performing in French, even though on some occasions, the rest of the cast were performing in English. In Paris, she had revived the play in 1899 to inaugurate the Théâtre Sarah Bernhardt where it ran for 57 nights and starred in another major Parisian revival in 1909, six months to the day after Sardou's death.

La Tosca had its US premiere within four months of its Paris opening, performed in English translation with Fanny Davenport in the title role and her husband, Willet Melbourne MacDowell, as Cavaradossi. The "Davenport Tosca" opened in New York City on 3 March 1888 and inaugurated the luxurious new Broadway Theatre on 41st Street. Davenport had previously bought the rights to the American premiere of Sardou's Féodora, and had made a fortune from it. She bought the rights to the American premiere of La Tosca for 100,0000 francs, before it had even premiered in Paris. As had happened at the Paris premiere, a charge of plagiarism was soon brought. Maurice Barrymore claimed that his 1884 play, Nadjezda, had been plagiarised by Sardou and sought an injunction to stop Davenport putting on further performances of La Tosca. According to Barrymore, he had given a copy of his play to Sarah Bernhardt in 1885, and she had then given it to Sardou. In affidavits read out in court Bernhardt said that she had never seen the play and knew nothing about it, and Sardou said that preliminary material for the play had been in his desk for fifteen years. In fact, Nadjezda's only resemblance to La Tosca comes from the unholy bargain the heroine makes to save her husband's life, similar to that of Tosca and Baron Scarpia. As Sardou pointed out in his affidavit, this plot device is a common one and had been notably used by Shakespeare in Measure for Measure. Davenport herself was in the courtroom on 27 April 1888 when the judge found in her favour. Following the New York run, she toured the play throughout the US with her company.

Tosca remained in Davenport's repertoire until the end of her career. After her death in 1898, her husband continued to tour the play with Blanche Walsh in the title role. Other prominent actresses who portrayed Floria Tosca in the play's heyday were the British actresses Fanny Bernard-Beere who performed the role in English at London's Garrick Theatre in 1889 and Ethel Irving who was still playing the role in 1920; the American actress Cora Urquhart Potter who toured the play in Australia and New Zealand; and the Italian actresses, Teresa Boetti Valvassura and Italia Vivanti (a cousin of Eleonora Duse). After the mid-1920s, revivals of the play became increasingly sporadic. It was performed in Canada by La Comédie de Montréal in 1941 starring Sita Riddez, and an English version adapted by Norman Ginsbury was broadcast on the BBC Home Service in 1958, but by then the play itself had completely disappeared from the standard theatrical repertoire.

Reception
Considered by Jerome Hart to be the most emotional of all Sardou's plays, La Tosca's critical reception was in sharp contrast to that of the opening night audience. The Parisian critics roundly attacked the play with Francisque Sarcey calling it a "pantomime", as did Jules Lemaître. Jules Favre writing in Les Annales politiques et littéraires called it a "vulgar piece, without intrigue, without characters, without morals". The New York Times correspondent reported the play's resounding success with the audience, but like many commentators of the day, including Favre, largely attributed it to Sarah Bernhardt's powerful performance, noting that:

"There is not much of play, a mere outline at best, made to fit like a glove the talent and personality of Bernhardt who is all and everything, but who should or could complain? The interest never slackens; there is enough dialogue and apropos to keep both gratification and amusement entertained, and the story enobles itself magically in the hands of the greatest living actress."

Writing from the perspective of the late 20th century, Nicassio agrees that Bernhardt's performance as a character essentially like herself, a celebrated, amorous, and temperamental diva, was undoubtedly a key factor in the play's success with the Paris audience. However, she cites other factors which also played a part: the "exotic" Italian setting with sumptuous sets and costumes, the play's anti-clerical themes, and a plot glorifying the Bonapartists as the 100th anniversary of the French Revolution approached.

Following the London premiere in 1888, Cecil Howard wrote that the play was even more popular there than it had been in Paris. Like several critics describing the Paris premiere, he devoted a large part of his review to Bernhardt's performance, which he said held the audience "breathless and rapt", but he had little admiration for Sardou's drama:

"As to the play itself, I will only add that it is offensive in its morals, corrupt in its teaching, and revolting in its brutality, and yet everyone who admires acting is bound to see it."

The "unchaste" behaviour of the heroine and the violence and brutality depicted in the play, although relatively mild by modern standards, disturbed not only critics at the time, but also some play-goers. The audience's reaction to Tosca's suicide at the American premiere caused Fanny Davenport to change the ending in subsequent performances with the firing squad taking aim at Tosca while she grieves over Cavaradossi's lifeless body, an ending also used by Sarah Bernhardt when she performed the play in Fort Worth, Texas in 1892. William Winter went so far as to warn American women that La Tosca contained scenes which were "not only shocking to the nervous system and grossly offensive to persons of true sensibility, but which might inflict irreparable injury on persons yet unborn." Several early critics, including Arthur Bingham Walkley and Jules Lemaître, wrote at length on Scarpia's graphic description of Cavaradossi's torture and the sound of his off-stage screams in Act 3, which they considered both gratuitously violent and inartistic. However, this was not a view shared by Oscar Wilde, who found the torture scene moving in its depiction of "a terrible human tragedy". George Bernard Shaw intensely disliked all of Sardou's work, and not surprisingly characterised La Tosca, which he saw in London in 1890, as a "clumsily constructed, empty-headed turnip ghost of a cheap shocker", while presciently suggesting that it would make a good opera.

Despite the views of the critics, La Tosca proved to be phenomenally successful. It ultimately had 3000 performances in France alone, played in theatres all over the world for thirty years, and netted Sardou 500,000 francs. Sarah Bernhardt's costumes brought Empire silhouette dresses back into style, and the long walking stick she carried in Act 1 became a new fashion accessory. Both a leopard in a famous New York menagerie and an American race horse were named in honour of the play's heroine, as were numerous dishes, several of them created by the French chef, Auguste Escoffier, a devotee of Bernhardt.

Adaptations
The most famous adaptation of La Tosca was Giacomo Puccini's Italian opera Tosca which premiered in Rome on 14 January 1900 with Hariclea Darclée in the title role and went on to successful premieres in London, New York, and Paris. The Paris premiere at the Opéra-Comique in 1903 was performed in a French translation by Paul Ferrier with Sardou himself taking charge of the rehearsals. Unlike Sardou's play, Puccini's opera has achieved an enduring popularity. More than 100 years after its premiere, Tosca ranks sixth in the list of most frequently performed operas worldwide, and has over 100 commercial recordings as well as several film versions (see Tosca discography). Puccini had seen La Tosca in Italy when Bernhardt toured the play there and asked his publisher, Giulio Ricordi, to negotiate with Sardou for the adaptation rights. Before Puccini obtained the rights, the composers Alberto Franchetti and Giuseppe Verdi had both expressed interest in turning La Tosca into an opera, although Verdi thought the ending had to be changed. Puccini's librettists, Luigi Illica and Giuseppe Giacosa, likewise tried (unsuccessfully) to convince Sardou to accept a new ending, with Tosca going mad rather than committing suicide. The Sardou ending stayed, but Illica and Giacosa did make several significant changes to the play, primarily to tighten the action.

Earlier, La Tosca had been adapted into an English novel by Arthur D. Hall in 1888, and had two adaptations for traditional Japanese theatre, both performed in 1891. In the Japanese adaptations, the famed story-teller, San'yūtei Enchō, set the work during the period of the 1837 rebellion by Ōshio Heihachirō, while Fukuchi Gen'ichirō adapted the play for Kabuki theatre. There were at least four silent film adaptations. A hand-coloured version starring Sarah Bernhardt was made in 1906 by Le Film d'Art, a French film company run by André Calmettes and Charles le Bargy. Bernhardt was so displeased with her performance that she refused to allow its release and tried to buy up and destroy all the negatives. Le Bargy and Calmettes then re-filmed the work, this time with Cécile Sorel as Tosca, and released it in 1908. The Bernhardt version re-surfaced and was released in 1912 by Universal Pictures. There was also a 1918 version by Paramount Pictures with Pauline Frederick as Tosca. Only fragments remain of the Italian film made the same year starring Francesca Bertini. Later films tended to be adaptations of Puccini's opera rather than Sardou's play with the notable exception of Carl Koch's 1941 Italian film Tosca starring Imperio Argentina as Tosca and Rossano Brazzi as Cavaradossi. Jean Renoir originally worked with Koch on the adaptation, but had to leave Italy at the outbreak of World War II. The film was released in the US in 1947 as The Story of Tosca.

Shortly after the first London performances of La Tosca, Francis Burnand and the composer Florian Pascal wrote a musical parody of the play entitled Tra-la-la Tosca or The High-Toned Soprano and the Villain Bass. In their burlesque version, Tosca murders Scarpia in the "Cafe Romano allo Strando", stabbing him with a huge rolled-up restaurant bill and then places one of the dish covers over his face. Cavaradossi, instead, is executed by a phalanx of photographers. The show premiered at London's Royalty Theatre in January 1890 and ran for 45 performances, with the critic Cecil Howard pronouncing it one of Burnand's finest efforts. Burnand had previously parodied Sardou's Féodora as Stage-Doora (1883) and Théodora as The O'Dora (1885), both of which ran at Toole's Theatre in London. In 2004, Lucio Dalla composed an Italian musical, Tosca, Amore Disperato (Tosca, Desperate Love), based largely on the structure of Puccini's opera, but with elements from Sardou's play. The setting was updated to modern times with costumes by Giorgio Armani. Tosca, Amore Disperato continues to be performed in Italy and was broadcast on RAI television in June 2010.

Differences between Sardou's play and Puccini's opera
The number of characters is sharply reduced in the opera, and the work shortened to three acts, leaving out much of the political motivations of the protagonists. In the opera, Angelotti and Cavaradossi already know each other. In the play, they had never met before, thus allowing considerable scope to explain their histories and backgrounds to each other. The roles of Tosca's maid and Cavaradossi's two servants were eliminated as were most of the characters in Act 2, although some of them such as the Marquis Attavanti and Queen Maria Carolina are alluded to in the opera. The gathering at the Farnese Palace in the presence of Queen Maria Carolina, Act 2 of the play, was eliminated. The setting of Act 2 and the events of Acts 3 and 4 in the play were then combined into the second act of the opera, which involved several significant changes.

Unlike the play, Scarpia shows Tosca the Marquise Attavanti's fan in Act 1, where Puccini's librettists contrive to have her return to the church following the departure of Angelotti and Cavaradossi. In the opera, both Cavaradossi's interrogation and torture and Scarpia's subsequent murder take place in the Farnese Palace. In the play, Cavaradossi's interrogation is set at his country house, where he was captured, while Scarpia's murder takes place at his apartment in the Castel Sant'Angelo. The news of the Austrian defeat at Marengo which formed the climax of Act 2 in La Tosca does not emerge in the opera until after Cavaradossi has been captured and tortured. Thus Scarpia is able to listen to Tosca's uninterrupted performance of the cantata (heard in a distant room of the palace).

Early audiences (especially in the United States and Britain) sometimes balked at the realism in Sardou's play, especially Cavaradossi's screams while he is being tortured off-stage. In Puccini's version, his screams are likewise heard by the audience. However his death by firing squad is even more explicit, occurring on stage in full view of the audience, rather than off stage as in the play. Tosca's final words before committing suicide in the play are addressed to Spoletta and his men. When he vows to send her to join her lover, she cries "J'y vais, canailles!" ("I am going, scoundrels!"). In the opera, her final words are addressed to Scarpia: "O Scarpia, avanti a Dio!" ("O Scarpia, [we meet] before God!"). The opera also gives Cavaradossi a soliloquy in the final act, "E lucevan le stelle" ("And the stars were shining"), in which he reflects on his past happiness with Tosca and his impending death. Other relatively minor changes include Puccini's addition of a singing shepherd boy as Cavaradossi awaits his execution and a change of the church in Act 1 from Sardou's Sant'Andrea al Quirinale to Puccini's Sant'Andrea della Valle. The latter actually has a potential hiding place for Angelotti. Its Barberini chapel incorporates a shallow chamber separated from the main part of the chapel by a grille.

title: La Tosca


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Bohai Bay

Bohai Bay is one of the three major bays of the Bohai Sea, the northwestern and innermost gulf of the Yellow Sea. It is bounded by the coastlines of eastern Hebei province (Tangshan and Cangzhou), Tianjin municipality and northern Shandong province (Binzhou and Dongying) south of the Daqing River estuary (which is an old mouth of Luan River in Laoting County) and north of the Yellow River estuary. It is the most southerly water in the northern hemisphere where sea ice can form.

The Bohai Bay is the drainage destination of the Hai River and 15 other rivers. Due to these rivers' muddy runoff, the bay used to be a highly silty water body, but extensive damming of the various river systems has greatly diminished siltage. Nevertheless, the Bohai Bay in effect concentrates the runoff of the whole eastern North China Plain, and the Bay is an intensely polluted body of water. Reduced silt deposition and sea level rise are causing problems with sea encroachment in some coastal areas. Fisheries were traditionally some of the richest in China, fed by enormous sediment runoff and extensive shallows to serve as hatcheries. Pollution, eutrophication, habitat destruction caused by land reclamation, and intense overfishing have resulted in a collapse of stocks, and a decline of trawl catch per unit of effort (CPUE) from 138.8 kg/net.hr to 11.2 kg/net.hr from 1959 to 1998.

The Bohai Bay is ringed by several major ports: the Port of Tianjin, the large Port of Tangshan itself which consists of three ports (Caofeidian, Jingtang and Fennan), and the Port of Huanghua, making the Bay into a very crowded waterway. Land reclamation in Tianjin and in Caofeidian have greatly changed the littoral zone, and destroyed much of the area's wetlands. Land reclamation has also affected migratory birds.

As is the case of most of the Bohai Sea, the Bohai Bay is rich in hydrocarbon deposits and has several active offshore oil fields. Jidong Nanpu contains 7500000000 oilbbl, while the bay as a whole is estimated to contain 146 Goilbbl. On June 4, 2011, a large oil spill occurred related to the China National Offshore Oil Corporation.

title: Bohai Bay


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Tubular bells

Tubular bells (also known as chimes) are musical instruments in the percussion family. Their sound resembles that of church bells, carillons, or a bell tower; the original tubular bells were made to duplicate the sound of church bells within an ensemble. Each bell is a metal tube, 30 - 38 mm in diameter, tuned by altering its length. Its standard range is C4–F5, though many professional instruments reach G5. Tubular bells are often replaced by studio chimes, which are a smaller and usually less expensive instrument. Studio chimes are similar in appearance to tubular bells, but each bell has a smaller diameter than the corresponding bell on tubular bells.

Tubular bells are sometimes struck on the top edge of the tube with a rawhide- or plastic-headed hammer. Often, a sustain pedal will be attached to allow extended ringing of the bells. They can also be bowed at the bottom of the tube to produce a very loud, very high-pitched overtone.

The tubes provide a purer tone than solid cylindrical chimes, such as those on a mark tree.

Chimes are often found in orchestral and concert band repertoire. It rarely plays melody, instead being used most often as a color to add to the ensemble sound. It does have solos occasionally, often depicting church bells.

In tubular bells, modes 4, 5, and 6 appear to determine the strike tone and have frequencies in the ratios 92:112:132, or 81:121:169, "which are close enough to the ratios 2:3:4 for the ear to consider them nearly harmonic and to use them as a basis for establishing a virtual pitch". The perceived "strike pitch" is thus an octave below the fourth mode (i.e., the missing "1" in the above series).

Classical music
Tubular bells first appeared between 1860 and 1870 in Paris. The Englishman John Harrington patented tubular bells made of bronze. Arthur Sullivan may have been the first composer to score for tubular bells in the orchestra, in 1886. In the early 20th century tubular bells were also incorporated into theater organs to produce effects.

Tubular bells as a substitute for church bells were first used by Giuseppe Verdi in his operas Il trovatore (1853) and Un ballo in maschera (1859) and by Giacomo Puccini in Tosca (1900).

Passages in classical music:

 * Giuseppe Verdi – Rigoletto (1851)
 * Giuseppe Verdi – Il trovatore (1853)
 * Giuseppe Verdi – Un ballo in maschera (1859)
 * Modest Mussorgsky – Boris Godunov (1869, 1872, 1874)
 * Pyotr Ilyich Tchaikovsky – 1812 Overture (1880)
 * Pietro Mascagni – Cavalleria rusticana (1890)
 * Ruggero Leoncavallo – The Bajazzo (1892)
 * Gustav Mahler – Symphony No. 2 (1895)
 * Giacomo Puccini – Tosca (1900)
 * Alexander Scriabin – Le Poème de l'extase (1908)
 * Anton Webern – Six Pieces for large orchestra (1909–10, revised 1928)
 * Claude Debussy – Ibéria (1910)
 * Gustav Holst – The Planets (1914–16)
 * Giacomo Puccini – Turandot (1926)
 * Edgard Varèse – Ionisation (1931)
 * Richard Strauss – Die schweigsame Frau (1935)
 * Paul Hindemith – Symphonic Metamorphosis of Themes by Carl Maria von Weber (1944)
 * Benjamin Britten – Albert Herring (1945)
 * Aaron Copland – Symphony No. 3 (1946)
 * Olivier Messiaen – Turangalîla-symphonie (1946–48)
 * Carl Orff – Antigonae (1949)
 * Dmitri Shostakovich – Symphony No. 11 (1957)
 * Olivier Messiaen – Chronochromie (1959–60)
 * Arvo Pärt – Cantus in Memoriam Benjamin Britten (1977)
 * David Stanhope – Folksongs for Band, Suite no. 3 (1991, revised 2016)
 * Daron Hagen – Shining Brow (1993)

In popular music
Multi-instrumentalist Mike Oldfield named his first album Tubular Bells, best known for providing the musical theme to The Exorcist film (1973). At the beginning of his solo symphony recording project in 1972, Oldfield discovered a set of tubular bells at The Manor Studio in Oxfordshire, England, and asked new owner Richard Branson for permission to use them. The chimes were played by Oldfield on parts of the album, and they supplied the album name. Branson's company Virgin Records benefitted tremendously as Oldfield's album sold millions of copies; this also contributed to Branson's personal wealth and industry leverage.

Other uses
Tubular bells can be used as church bells, such as at St. Alban's Anglican Church in Copenhagen, Denmark. These were donated by Charles, Prince of Wales.

Tubular bells are also used in longcase clocks, particularly because they produce a louder sound than gongs and regular chime-rods and therefore could be heard more easily.

title: Tubular bells


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Vorticity equation

The vorticity equation of fluid dynamics describes the evolution of the vorticity $ω$ of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:$$where $D⁄Dt$ is the material derivative operator, $u$ is the flow velocity, $ρ$ is the local fluid density, $p$ is the local pressure, $τ$ is the viscous stress tensor and $B$ represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.

The equation is valid in the absence of any concentrated torques and line forces for a compressible, Newtonian fluid. In the case of incompressible flow (i.e., low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation:
 * $$\frac{D\boldsymbol\omega}{Dt} = \left(\boldsymbol{\omega} \cdot \nabla\right) \mathbf{u} + \nu \nabla^2 \boldsymbol{\omega}$$

where $ν$ is the kinematic viscosity and $$\nabla^{2}$$ is the Laplace operator. Under the further assumption of two-dimensional flow, the equation simplifies to:
 * $$\frac{D\boldsymbol\omega}{Dt} = \nu \nabla^2 \boldsymbol{\omega}$$

Physical interpretation
\frac{\partial\rho}{\partial t} + \nabla \cdot\left(\rho \mathbf u\right) &= 0 \\ \Longleftrightarrow \nabla \cdot \mathbf{u} &= -\frac{1}{\rho}\frac{d\rho}{dt} = \frac{1}{v}\frac{dv}{dt} \end{align}$$ where $Dω⁄Dt$ is the specific volume of the fluid element. One can think of $ω$ as a measure of flow compressibility. Sometimes the negative sign is included in the term.
 * The term $∂ω⁄∂t$ on the left-hand side is the material derivative of the vorticity vector $(u ∙ ∇)ω$. It describes the rate of change of vorticity of the moving fluid particle. This change can be attributed to unsteadiness in the flow ($(ω ∙ ∇) u$, the unsteady term) or due to the motion of the fluid particle as it moves from one point to another ($(ω ∙ ∇) u$, the convection term).
 * The term $ω ∙ ∇$ on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. Note that $∇u$ is a vector quantity, as $ω(∇ ∙ u)$ is a scalar differential operator, while $v = 1⁄ρ$ is a nine-element tensor quantity.
 * The term $∇ ∙ u$ describes stretching of vorticity due to flow compressibility. It follows from the Navier-Stokes equation for continuity, namely $$\begin{align}
 * The term $1⁄ρ^{2}∇ρ × ∇p$ is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces.
 * The term $∇ × (∇ ∙ τ⁄ρ)$, accounts for the diffusion of vorticity due to the viscous effects.
 * The term $∇ × B$ provides for changes due to external body forces. These are forces that are spread over a three-dimensional region of the fluid, such as gravity or electromagnetic forces. (As opposed to forces that act only over a surface (like drag on a wall) or a line (like surface tension around a meniscus).

Simplifications

 * In case of conservative body forces, $∇ × B = 0$.
 * For a barotropic fluid, $∇ρ × ∇p = 0$. This is also true for a constant density fluid (including incompressible fluid) where $∇ρ = 0$. Note that this is not the same as an incompressible flow, for which the barotropic term cannot be neglected.
 * For inviscid fluids, the viscosity tensor $τ$ is zero.

Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to
 * $$\frac{d}{dt} \left( \frac{\boldsymbol \omega}{\rho} \right) = \left( \frac{\boldsymbol\omega}{\rho} \right) \cdot \nabla \mathbf u $$

Alternately, in case of incompressible, inviscid fluid with conservative body forces,
 * $$\frac{d \boldsymbol \omega}{dt} = \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla)\boldsymbol \omega = (\boldsymbol \omega \cdot \nabla) \mathbf u $$

For a brief review of additional cases and simplifications, see also. For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.

Derivation
The vorticity equation can be derived from the Navier–Stokes equation for the conservation of angular momentum. In the absence of any concentrated torques and line forces, one obtains:

 * $$\frac{D \mathbf{u}}{D t} = \frac{\partial \mathbf{u}}{\partial t} + \left(\mathbf{u} \cdot \nabla\right) \mathbf{u} = -\frac{1}{\rho} \nabla p + \frac{\nabla \cdot \tau}{\rho} + \frac{\mathbf{B}}{\rho}$$

Now, vorticity is defined as the curl of the flow velocity vector; taking the curl of momentum equation yields the desired equation. The following identities are useful in derivation of the equation:
 * $$\begin{align}

\boldsymbol{\omega} &= \nabla \times \mathbf{u} \\ \left(\mathbf{u} \cdot \nabla\right)\mathbf{u} &= \nabla \left(\frac{1}{2}\mathbf{u} \cdot \mathbf{u}\right) - \mathbf{u} \times \boldsymbol\omega \\ \nabla \times \left(\mathbf{u} \times \boldsymbol{\omega} \right) &= -\boldsymbol{\omega} \left(\nabla \cdot \mathbf{u}\right) + \left(\boldsymbol{\omega} \cdot \nabla\right) \mathbf{u} - \left(\mathbf{u} \cdot \nabla\right) \boldsymbol{\omega} \\[4pt] \nabla \cdot \boldsymbol{\omega} &= 0 \\[4pt] \nabla \times \nabla \phi &= 0 \end{align}$$

where $$\phi$$ is any scalar field.

Tensor notation
The vorticity equation can be expressed in tensor notation using Einstein's summation convention and the Levi-Civita symbol $e_{ijk}$:

 * $$\begin{align}

\frac{D\omega_i}{Dt} &= \frac{\partial \omega_i}{\partial t} + v_j \frac{\partial \omega_i}{\partial x_j} \\ &= \omega_j \frac{\partial v_i}{\partial x_j} - \omega_i \frac{\partial v_j}{\partial x_j} + e_{ijk}\frac{1}{\rho^2}\frac{\partial \rho}{\partial x_j}\frac{\partial p}{\partial x_k} + e_{ijk}\frac{\partial}{\partial x_j}\left(\frac{1}{\rho}\frac{\partial \tau_{km}}{\partial x_m}\right) + e_{ijk}\frac{\partial B_k }{\partial x_j} \end{align}$$

Atmospheric sciences
In the atmospheric sciences, the vorticity equation can be stated in terms of the absolute vorticity of air with respect to an inertial frame, or of the vorticity with respect to the rotation of the Earth. The absolute version is

\frac{d\eta}{dt} = -\eta\nabla_\text{h} \cdot \mathbf{v}_\text{h} - \left(   \frac{\partial w}{\partial x} \frac{\partial v}{\partial z} -    \frac{\partial w}{\partial y} \frac{\partial u}{\partial z}  \right) - \frac{1}{\rho^2}\mathbf{k} \cdot \left(\nabla_\text{h} p \times \nabla_\text{h}\rho\right) $$

Here, $η$ is the polar ($z$) component of the vorticity, $ρ$ is the atmospheric density, $u$, $v$, and w are the components of wind velocity, and $∇_{h}$ is the 2-dimensional (i.e. horizontal-component-only) del.

title: Vorticity equation


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Mulled wine

Mulled wine, also known as spiced wine, is an alcoholic drink usually made with red wine, along with various mulling spices and sometimes raisins, served hot or warm. It is a traditional drink during winter, especially around Christmas. It is usually served at Christmas markets in Europe, primarily in Germany, Czech Republic, Austria, Switzerland, Slovenia, Croatia, Hungary, Romania, Nordics, Baltics and eastern France. There are non-alcoholic versions of it. Vodka-spiked mulled wine can be found in Polish Christmas markets, where mulled wine is commonly used as a mixer.

Origins
The first record of wine being spiced and heated can be found in Plautus' play Curculio, written during the 2nd century BC. The Romans travelled across Europe, conquering much of it and trading with the rest. The legions brought wine and viticulture with them up to the Rhine and Danube rivers and to the Scottish border, along with their recipes.

The Forme of Cury, a medieval English cookery book from 1390, which mentioned mulled wine, says: "Pur fait Ypocras ..." grinding together cinnamon, ginger, galangal, cloves, long pepper, nutmeg, marjoram, cardamom, and grains of paradise ("spykenard de Spayn", rosemary may be substituted). This is mixed with red wine and sugar (form and quantity unstated).

Britain
Mulled wine is very popular and traditional in the United Kingdom at Christmas, and less commonly throughout winter. Mulled cider (and sometimes mulled ale, traditional yet no longer common) is also served, with a mulled apple juice as a non-alcoholic alternative.

In traditional culture

Over the years the recipe for mulled wine has evolved with the tastes and fashions of the time. One Victorian example of this is smoking bishop, mentioned by Charles Dickens but no longer drunk or known in modern culture. A more traditional recipe can be found in Mrs. Beeton's Book of Household Management at paragraph 1961 on page 929 to 930 of the revised edition dated 1869: "TO MULL WINE. INGREDIENTS.- To every pint of wine allow 1 large cupful of water, sugar, and spice to taste.

Mode.-In making preparations like the above, it is very difficult to give the exact proportions of ingredients like sugar and spice, as what quantity might suit one person would be to another quite distasteful. Boil the spice in the water until the flavour is extracted, then add the wine and sugar, and bring the whole to the boiling point, then serve with strips of crisp dry toast, or with biscuits. The spices usually used for mulled wine are cloves, grated nutmeg, and cinnamon or mace. Any kind of wine may be mulled, but port and claret are those usually selected for the purpose, and the latter requires a very large proportion of sugar. The vessel that the wine is boiled in must be delicately cleaned and should be kept exclusively for the purpose. Small tin warmers may be purchased for a trifle, which is more suitable than saucepans, as, if the latter are not scrupulously clean, they spoil the wine, by imparting to it a very disagreeable flavour. These warmers should be used for no other purpose."

In contemporary culture

In contemporary British culture, there is no specific recipe for mulled wine and the spices involved in its recipe. It is commonly a combination of orange, lemon, cinnamon, nutmeg, fennel seed (or star anise), cloves, cardamom, and ginger. The spices may be combined and boiled in a sugar syrup before red wine is added, heated, and served. Variations include adding brandy or ginger wine. A tea bag of spices can be added to the wine, which is heated along with slices of orange as a convenient alternative to a full recipe. Mulled wine is often served in small (200 ml) porcelain or glass mugs, sometimes with an orange slice garnish studded with cloves.

Mulled wine and ales infused with mulling spices are available in the UK in the winter months. Wassail punch is a warm mulled beer or cider drunk in winter in Victorian times.

Glühwein

Glühwein (roughly translated as "smouldering-wine", from the temperature the wine is heated to) is popular in German-speaking countries, the Germany-bordering Alsace region of France, and to a lesser extent Russia. It is a traditional beverage offered during the Christmas holidays. In Alsace Christmas markets, it is traditionally the only alcoholic beverage served. The oldest documented Glühwein tankard is attributed to Count John IV of Katzenelnbogen, a German nobleman who was the first grower of Riesling grapes. This gold-plated lockable silver tankard is dated to c. 1420.

Glühwein is usually prepared from red wine, heated and spiced with cinnamon sticks, cloves, star aniseed, orange, sugar and at times vanilla pods. It is sometimes drunk mit Schuss (with a shot), which means that rum or some other liquor has been added. Fruit wines, such as blueberry wine and cherry wine, are occasionally used instead of grape wine in some parts of Germany. A variation of Glühwein is made with white wine, but it is less popular than its red counterpart. For children, the non-alcoholic Kinderpunsch is offered on Christmas markets, which is a punch with similar spices.

Another popular variant of Glühwein in Germany is Feuerzangenbowle. It shares the same recipe, but for this drink a rum-soaked sugarloaf is set on fire and allowed to drip into the wine. Polish type of Glühwein is called Grzaniec. It's similar to the German one with one exception - it is sweetened usually with honey instead of sugar.

Nordic glögg

Glögg, gløgg, glögi and similar words are the terms used for mulled wine in the Nordic countries and Estonia (sometimes spelled as glog or glug). It is spelled gløgg in Norwegian, Danish and Faroese, glögg in Swedish and Icelandic and glögi in Finnish and Estonian. In Denmark, Norway, Sweden and Finland gløgg or glögg is often drunk at Christmas events.

Non-alcoholic and alcoholic versions of glögg can be bought ready-made or prepared with fruit juices instead of wine. The main ingredients of alcoholic glögg are red wine, sugar, spices such as cinnamon, cardamom, ginger, cloves, and bitter orange, and optionally also stronger spirits such as vodka, akvavit, rum or brandy. Throughout Scandinavia, glögg spice extract and ready-mixed spices can be purchased in grocery stores. To prepare glögg, spices or spice extract are mixed into the wine, which is then heated to 60-70 °C. When preparing homemade glögg using spices, the hot mixture is allowed to infuse for at least an hour, often longer, and then reheated before serving. Ready-made wine glögg, as well as low- or non-alcoholic varieties, is normally sold at Systembolaget in Sweden, and in Alko in Finland, ready to heat and serve, and not in concentrate or extract form. Glögg is generally served with raisins, dried cloves, blanched almonds and ginger biscuits (gingerbread cookies), and is a popular hot drink during the Christmas season.

In Sweden, ginger bread and lussebullar (also called lussekatter), a type of sweet bun with saffron and raisins, are typically served on December 13 to celebrate Saint Lucia's Day. It is also traditionally served at the julbord, the Christmas version of the classic, Swedish buffet smörgåsbord. In Denmark, gløgg pairings typically include æbleskiver sprinkled with powdered sugar and accompanied with strawberry marmalade. In Norway, gløgg is paired with risengrynsgrøt (rice pudding). In such cases, the word graut-/grøtfest is more precise, taking the name from the rice pudding which is served as a course. Typically, gløgg is drunk before eating the rice pudding, which is often served with cold, red cordial (saus).

Glögg recipes vary widely; variations with white wine or sweet wine such as Port or Madeira, or spirits such as brandy, schnapps or whisky are also popular, either added directly or by soaking the raisins and spice in them before preparing the drink (often resulting in a notable increase in the alcohol contents). Glögg can also be made without alcohol by replacing the wine with fruit or berry juices (often blackcurrant) or by boiling the glögg to evaporate the alcohol. Glögg is similar in taste to modern Wassail or mulled cider.

Other countries

In Croatia, Slovenia, Bosnia and Herzegovina, Montenegro and Serbia, kuhano vino or kuvano vino (кувано вино; lit. "cooked wine"), is made from red or white wine and various combinations of nutmeg, cloves, cinnamon, aniseed sugar/honey or orange zest, often served with slices of orange or lemon. In North Macedonia, it is called vareno vino (варено вино, "boiled wine") or greeno vino (греено вино, "heated wine") and is usually served in late autumn or winter. It is made of red wine, usually from the Tikvešh region, combined with cinnamon and sugar or honey. The wine, heated in combination with pepper, is used as a folk remedy for cold and flu.

In the south and southeast of Brazil, it is called vinho quente or quentão, from the same Portuguese drink. It is typically made with red wine, cinnamon sticks and cloves. Cachaça may be added to increase the alcohol content. It is served as part of the Festa Junina, celebrated during winter in the month of June.

In Bulgaria, it is called greyano vino (греяно вино; "heated wine"), and consists of red wine, honey and peppercorn. Sometimes apples or citrus fruits, such as lemon or oranges, can be added.

In Chile, it is called candola in the south and vino navega'o in the north (navegado; lit. "sailor"; "navigated" is considered a hypercorrection). Navega'o is a hot drink made from red wine, orange peels, cinnamon sticks, cloves, and sugar. Although being considered a southern Chile beverage, it is served throughout the country. Many people regard it as a winter drink. Saint John's Eve (Spanish tradition which replaced Wetripantru, the Mapuche New Year's Day that coincides with the Winter Solstice in the southern hemisphere—Midsummer in the northern hemisphere) on the evening of 23 June would be, for example, a good time to drink navega'o.

In the Czech Republic, mulled wine is called svařené víno ("boiled wine"), colloquially svařák. In Slovakia, mulled wine is called varené víno ("boiled wine"), and is usually served during the Christmas season.

In France, vin chaud ('hot wine') typically consists of red wine mixed with honey, cinnamon, and orange. It must not be too sweet. Beverage noted in the Alps for winter sports. In Geneva, Switzerland, vin chaud is consumed during the festivities surrounding L'Escalade.

In Hungary, forralt bor ("boiled wine") is typically made from the country's popular Egri Bikavér and spiced with cinnamon, sugar and cloves. Sometimes Amaretto is added for extra taste.

In Italy, mulled wine is typical in the northern part of the country and is called vin brülé (from the French vin brûlé, "burnt wine", though the expression is not used in France).

In Latvia, it is called karstvīns ("hot wine"). When out of wine, it is prepared using grape (or currant) juice and Riga Black Balsam. In Lithuania, it is called glintveinas or karštas vynas ("hot wine").

In Romania, vin fiert is typically made with cinnamon, sugar and oranges. Red wine is the most common type but white wine is also used. In Moldova vin fiert is made from red wine with black pepper and honey.

In the Netherlands and Belgium, the drink is known as bisschopswijn ('bishop's wine'). Bisschopswijn is drunk during the Sinterklaas holidays.

In Poland, grzane wino  ("heated wine"), or grzaniec in highlander dialect, is very similar to the Czech variant, especially in the southern regions. There is also a similar method for preparing mulled beer, or grzane piwo, which is popular with Belgian beers because of the sweet flavor of that particular type of beer, which uses the same spices as mulled wine and is heated. Vodka-spiked mulled wine can be found in Polish Christmas markets, where mulled wine is used as a mixer.

In Portugal, mainly in the Douro and Minho Provinces, it is called vinho quente and made with Madeira wine and port wine, in the Porto region Porto Quente is more popular.

In Russia and Ukraine, glintveyn (глинтвейн) is a popular drink during winters and has same recipe as the German Glühwein. Additionally, the traditional Russian winter herbal drink sbiten, although usually a non-alcoholic tisane made with hot water, may also be made with red wine replacing some or all of the water.

In popular culture
In the 1946 Christmas movie It's a Wonderful Life, after the 293 year old guardian angel Clarence Odbody (traveling with George Bailey to an alternate world if George had not been born) goes to a rough bar. And, not knowing what to order since he had not been to a bar in centuries, decides to order, "I was just thinking of a flaming rum punch. No, it’s not cold enough for that. Not nearly cold enough…Wait a minute…wait a minute…Mulled wine, heavy on the cinnamon and light on the cloves. Off with you, me lad, and be lively!

title: Mulled wine