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One of the most important elements of a successful wood flooring installation job is the application of the finish. A perfect installation can be distorted by a bad finish job, and a mediocre installation can often be saved by a good finish job. The customer's expectations of what the finish is supposed to do can trump a good or a bad finish job.
Today's finishes continue to evolve, and as professionals in this trade, we are expected to understand the nuances of every finish in order to create what our customers ask us to produce. Fortunately (or unfortunately, depending on how you look at it), all finishes and finish systems are not identical in their application or their performance.
Finish Wear Discrepancies: When the performance of the surface coat of finish is not meeting expectations.
Finish selection used on the floor is not properly matched to the use of the floor.
Maintenance is a common culprit. Poor maintenance practices may include failure to remove grit from the floor's surface fully, missing or unsuitable area rugs, walk-off mats, non-slip pads, worn or inadequate floor protection under furniture, pet nails, vacuuming with the carpet setting (beater bar) on the wood floor surface, and most common is using too much or the wrong cleaning product (including steam cleaners) to clean the floor.
The performance of a new wood floor installed in a new construction jobsite often comes into question. Other than the normal heavy construction traffic, other causes for finish wear on the jobsite include inadequate or no floor protection used during construction. Another cause is when floor protection is applied, grit, dirt, and construction debris under the floor protection causes finish wear. Finally, the maintenance company improperly cleaning the floors.
Finish-related issues may include any of the following: improper finish application, inadequate jobsite conditions affecting the drying and curing of the finish, improperly mixed finish, coating over a not-yet cured sealer or finish coat, a bad batch or finish used at or beyond its expiration/best-use date, using finish that has been exposed to extreme temperatures, floor use before the finish has fully cured, floors covered/protected (area rugs, floor protection) too soon after finish application, and inadequately cured site-cured UV finishes.
The sand job can also play a part, including inconsistent sanding procedures accompanied by an inadequate abrasive grit selection for the floor being sanded and/or the finish being used. This often results in high points/peaks coated with less finish than the low points/valleys.
For factory-finished flooring, the high points/peaks of sculpted floors, distressed floors, overwood/underwood, etc., wear through faster than low points/valleys, inadequately cured or improperly applied factory finishes.
Scuffs: Marks or abrasions left on a wood floor surface or finish.
Most scuffs are maintenance-related. Some of the sources may include grit or debris rubbed across the floor's surface, improper maintenance products, which can leave a residue or film on the surface of the wood floor, often have a tendency to scuff or become marred, and hard, stiff soles on shoes, boots, sports cleats, high heels that are not specifically designed to be "non-marking soles." These normally stem back to the customer expectations of the floor.
Scuffing can also be directly related to the finish. Some of the causes may include early foot traffic or use of the floor on uncured finishes, improperly mixed, applied, or stored finish, expired finish, or a bad batch of finish. UV-cured finishes can also scuff if it was improperly cured.
Impact Damage: A depression, mark, or wound on the surface of the finished wood flooring. All species and types of wood flooring and wood flooring finishes are prone to impact damage resulting in scratches, dents, and gouges. Common causes for scratches, indentations, and gouges on wood flooring may include high heels, sports cleats, pet nails, appliances such as refrigerators or stoves, furniture legs, piano feet, dropped objects, shipping or packaging damage, installation damage, and construction damage. Proper identification of each type of damage may help in the assessment.
Scratches are caused by applied horizontal and vertical forces and are typically only in the finish coats. A fingernail can't feel the ridge. Deep scratches are typically through the finish coats and have torn the wood fibers. Cross-grain scratches are deep scratches and are oriented perpendicular to the direction of the flooring, often crossing multiple board widths and have a higher probability of exhibiting wood fiber tear. Deep scratches, usually termed "gouges," are typically through the finish coats and have torn the wood fibers. As scratch damage increases in severity, so does the likelihood that the damage will also include denting.
Indentations are notches, grooves or depressions on the surface of the flooring. Indentations are caused by applied vertical and/or horizontal forces on the flooring surface. When only vertical forces have been applied (dropping an object) these depressions are typically isolated to the shape of the portion of the dropped object that contacted the floor. Indentations may not always be accompanied with finish tear.
Wood bruises originate from a previous force or trauma to the surface of the boards that is not normally noticeable during the installation or handling of the product. This damage can become exposed with an increase in moisture, such as with the introduction of water through water-popping, water-based finishes, water-based stains, or similar reactive conditioners.
Various rug types or backing materials that can leave an imprint in the floor finish system, or plasticizer migration from some non-slip backing pads used under area rugs can leave a waffle-like pattern etched into the finish system. Area rugs or non-slip area rug pads that are incompatible with the wood floor finish systems or have been placed over uncured finishes can leave discoloring stains, blushing of the finishes, or impressions in the finish as well as the wood.
Floor protection systems (corrugated cardboard, paper, pads, etc.) laid over the wood floor before the finish is fully cured (per manufacturer instructions) can leave imprints within and on the finish system. Floor protection systems with any sort of grit or debris caught between the protection and floor can leave impressions in the finished floor.
Factory-finished flooring packaging material has been known to transfer to the face of the flooring boards.
Finish Adhesion Failure: Occurs when there is a loss of film-finish adhesion to the flooring surface or between coating layers. Finish adhesion failure is broken into two categories: chipping/flaking finish and peeling finish.
Poor adhesion between coats of finish as a result of contamination, poor abrasion, improper preparation, incompatible finishes, or even using different types of finishes on subsequent coats.
With some more-brittle/less-elastic finishes used on some factory-finished floors, a gain or loss in moisture, and subsequent dimensional changes within the wood can result in finish chipping or shearing away from the wood.
Excessive finish on the edge of factory-finished flooring (also known as cornicing) can chip as the boards are being driven together during installation.
Use of steam mops to clean wood floors can cause the finish to chip away from the seams or voids in the wood.
Poor adhesion between coats of finish as a result of contamination, poor abrasion, improper preparation, incompatible finishes, finish applied over undried or insufficiently wiped stain, or even using different types of finishes on subsequent coats.
When finish peels from a previously finished floor that has been recoated. The causes may include improper preparation, inadequately cleaning the floor surface prior to recoat, poor abrasion in bevels, insufficient or incompatible finish and adhesion system selection, or incompatible finishes. NOTE: Factory-finished floors marketed to be stain resistant cannot be recoated. Stain resistant finishes such as Perfluorooctane sulfonate (Teflon/Scotchgard) infused finishes will not accept a new coat of finish.
Using tape on a finished floor that was not recommended to be used on wood floors; leaving the tape adhered to the wood floor for a longer time than the tape manufacturer's recommendation; applying the tape to a newly coated wood floor finish; and plasticizers in the tape that have created a stronger bond to the finish than the finish has to the previous coatings or the floor.
When drywall work has been scheduled after wood floor installation, the wood floor is susceptible to becoming exposed to drywall dust. Subsequent wetting/damp mopping of the surface will turn the drywall dust into a paste, which can be extremely difficult to remove from the exposed voids or characteristics of the flooring.
Silt, salt residue, or other fine dust particles can become embedded in the exposed voids or characteristics of the unprotected flooring.
Paint, flooring adhesive, soldering flux from the plumber, or other contaminants that have been spilled and subsequently cleaned from the surface can remain in the exposed voids or characteristics of the flooring.
Improper maintenance practices such as using products and cleaning systems not recommended by the flooring or finish manufacturer, or using an old or soiled cleaning mop head to clean the floor.
A Slippery Floor: Defined as the perception that the floor feels slippery. All wood floor finishes could be considered slippery with normal use. Most wood floor finishes are not usually considered to be slip-resistant, and some wood floor finishes feel more slippery than others.
A few things that can cause a floor to be perceived as being slippery include the type of footwear, a person's weight, stride length, gait, and speed that all can affect the degree to which one slides across any flooring surface.
Improper maintenance practices can also cause a slippery floor. Improper maintenance may include overspray from using cleaning products on fixtures on or near the wood floor, tracking of contaminants from one floor surface onto the wood floor, using cleaning products on the floor not recommended by the finish or flooring manufacturer, using old or soiled cleaning mop heads, or simply dust or debris on the floor surface.
Finish-related causes for slippery floors may include over-application of wax on a waxed floor (using too much wax) or adding non-manufacturer-recommended substances to the finish.


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text:
Report on Casting Process of Materials
Subject: Engineering, Science Topic: Report
Casting is a process by which a material is introduced into a mold while it is liquid, allowed to solidify in the shape inside the mold, and then removed producing a fabricated object, part, or casting. Casting is often used for creating one or more copys of an original piece of sculptural (three-dimensional) artwork. It is also used extensively in the automobile
One half of a bronze mold for casting a socketed spear head dated to the period 1400-1000 BC.
manufacture industry, such as the casting of engine blocks or cylinder heads, or vacuum-forming of plastics and in the lost core process. The process, particularly when performed with molten metals, is also called founding.
Casting may be used to form hot, liquid metals or meltable plastics (called thermoplastics), or various materials that cold set after mixing of components such as certain plastic resins (e.g. epoxy), water setting materials such as concrete or plaster, and materials that become liquid or paste when moist such as clay, which when dry enough to be rigid is removed from the mold, further dried, and fired in a kiln.
Casting may be used to create artistic sculptures
 1 Lost wax casting process
 2 Other casting processes used in creating artworks
 3 Casting in manufacturing
 3.1 Expendable mold casting
 3.1.1 Sand Casting
 3.1.2 Plaster casting (of metals)
 3.1.3 Plaster casting (of plaster or concrete)
 3.1.4 Shell molding
 3.1.5 Investment Casting
 3.2 Nonexpendable mold casting
 3.2.1 Permanent Mold Casting
 3.2.2 Die-Casting
 3.2.3 Centrifugal casting
 3.2.4 Continuous Casting
 4 Cooling rate
 5 Shrinkage
 5.1 Volumetric Shrinkage
 5.2 Linear Shrinkage
Lost wax casting process
The Lost Wax casting process is an ancient practice that is still in widespread use today. The process varies from foundry to foundry, but the steps which are usually used in casting small bronze sculptures in a modern bronze foundry are as follows:
1. Sculpting. An artist creates an original artwork from wax, clay, or another material. Wax and oil-based clay are often preferred because these materials retain their softness.
2. Mouldmaking. A mould is made of the original sculpture. Most moulds are at least two pieces, and a shim with keys is placed between the two halves during construction so that the mould can be put back together accurately. Most moulds of small sculptures are made from plaster, but can also be made of fiberglass or other materials. To preserve the fine details on the original artwork's surface, there is usually an inner mould made of latex or vinyl, which is supported by the plaster part of the mould. Usually, the original artwork is destroyed during the making and initial deconstruction of the plaster mould. This is because the originals are solid, and do not easily bend as the plaster mould is removed. Often long, thin pieces are cut off of the original and moulded separately. Sometimes, especially in the case of large original (such as life-size) sculptures, many moulds are needed to recreate the original sculpture.
3. Wax. Once the plaster and latex mould is finished, molten wax is poured into it and swished around until an even coating, usually about 1/4 inches think, covers the entire inner surface of the mould. This may be done in several layers.
4. Removal of wax. This new, hollow wax copy of the original artwork is removed from the mould. The artist may reuse the mould to make more wax copies, but wear and tear on the mould limit their number. For small bronze artworks, a common number of copies today is around 25.
5. Chasing. Each hollow wax copy is then "chased": a heated metal tool is used to rub out all the marks which show the "parting line" or "flashing" where the pieces of the mould came together. Wax pieces that were moulded separately can be heated and attached; foundries often use "registration marks" to indicate exactly where they go.
6. Spruing. Once the wax copy looks just like the original artwork, it is "sprued" with a treelike structure of wax that will eventually provide paths for molten bronze to flow. The carefully-planned spruing usually begins at the top with a wax "cup," which is attached by wax cylinders to various points on the wax copy.
7. Slurry. A "sprued" wax copy is dipped into a ceramic slurry, then into a mixture of powdered clay and sand. This is allowed to dry, and the process is repeated until a half-inch thick or thicker surface covers the entire piece. Only the inside of the cup is not coated, and the cup's flat top serves as the base upon which the piece stands during this process.
8. Burnout. The ceramic-coated piece is placed cup-down in a kiln, whose heat hardens the ceramic coatings into a shell, and the wax melts and runs out. The melted wax can be recovered and reused, although often it is simply combusted by the burnout process. Now all that remains of the original artwork is the negative space, formerly occupied by the wax, inside the hardened ceramic shell. The feeder and vent tubes and cup are now hollow, also.
9. Testing. The ceramic shell is allowed to cool, then is tested to see if water will flow through the feeder and vent tubes as necessary. Cracks or leaks can be patched with thick ceramic paste. To test the thickness, holes can be drilled into the shell, then patched.
10. Pouring. The shell is reheated in the kiln to harden the patches, then placed cup-upwards into a tub filled with sand. Bronze is melted in a crucible in a furnace, then poured carefully into the shell. If the shell were not hot, the temperature difference would shatter it. The bronze-filled shells are allowed to cool.
11. Release. The shell is hammered or sand-blasted away, releasing the rough bronze. The spruing, which are also faithfully recreated in metal, are cut off, to be reused in another casting.
12. Metal-chasing. Just as the wax copies were "chased," the bronze copies are worked until the telltale signs of casting are removed, and the sculptures again look like the original artwork. Pits left by air bubbles in the molten bronze are filled, and the stubs of spruing filed down and polished.
13. Patinating. The bronze is coloured to the artist's preference, using chemicals applied to heated or cooled metal. This colouring is called patina, and is often green, black, white or brownish to simulate the surfaces of ancient bronze sculptures. (Ancient bronzes gained their patinas from oxidisation and other effects of being on Earth for many years.) However, many artists prefer that their bronzes have brighter, paint-like colours. Patinas are generally less opaque than paint, which allows the lustre of the metal to show through. After the patina is applied, a coating of wax is usually applied to protect the surface. Some patinas change colour over time because of oxidisation, and the wax layer slows this down somewhat.
The lost-wax process can also be used with any material that can burn, melt, or evaporate to leave a mold cavity. Some automobile manufacturers use a lost-foam technique to make engine blocks. The model in this case is made of polystyrene foam, which is then placed into a casting flask, consisting of a cope and drag, which is then filled with casting sand. The foam supports the sand, allowing shapes to be made which would not be possible if the process had to rely on the sand alone to hold its shape. The metal is then poured in, and the heat of the metal vaporizes the foam as the metal enters the mold.
Other casting processes used in creating artworks
Sand-casting is mainly used for casting flat, relief-like sculptures. Aluminium is one material which is commonly used in sand-casting. The process starts with a tub filled with sand. The sand is wetted, and an object is pressed into the wet sand, or the sculptor uses his hands or tools to make the desired design in the sand, which is then dried. Molten aluminium is carefully poured into the depression and left to cool. Then the artist may choose to continue refining the object by "chasing" it or leave it with the roughened surface that is characteristic of sand-cast objects.
Cuttlefish casting using cuttlebone as a mould is a traditional casting method used by jewellers and silversmiths for small objects, especially in taking a copy from a metal original. The fine grain of the calcium carbonate cuttlebone offers good definition, although it imparts a characteristic surface texture to the cast.
Casting in manufacturing
Casting is a process by which a fluid melt is introduced into a mold, allowed to cool in the shape of the form, and then ejected to make a fabricated part or casing. Four main elements are required in the process of casting: pattern, mold, cores, and the part. The pattern, the original template from which the mold is prepared, creates a corresponding cavity in the casting material. Cores are used to produce tunnels or holes in the finished mold, and the part is the final output of the process.
Substitution is always a factor in deciding whether other techniques should be used instead of casting. Alternatives include parts that can be stamped out on a punch press or deep-drawn, items that can be manufactured by extrusion or by cold-bending, and parts that can be made from highly active metals.
The casting process is subdivided into two distinct subgroups: expendable and nonexpendable mold casting.
Expendable mold casting
Expendable mold casting is a generic classification that includes sand, plastic, shell, and investment (lost-wax technique) moldings. All of these involve the use of temporary and nonreusable molds, and need gravity to help force molten fluid into casting cavities.In this process the mould is used only once.
Sand casting requires a lead time of days for production at high output rates (1-20 pieces/hr-mold), and is unsurpassed for large-part production. Green (wet) sand has almost no part weight limit, whereas dry sand has a practical part mass limit of 2300-2700 kg. Minimum part weight ranges from 0.075-0.1 kg. Sand in most operations can be recycled many times and requires little additional input.
Preparation of the sand mold is fast and requires a pattern which can "stamp" out the casting template. Typically, sand casting is used for processing low-temperature steel and aluminium, magnesium, and nickel alloys. It is by far the oldest and best understood of all techniques. Consequently, automation may easily be adapted to the production process, somewhat less easily to the design and preparation of forms. These forms must satisfy exacting standards as they are the heart of the sand casting process – creating the most obvious necessity for human control.
Plaster casting (of metals)
Plaster casting is similar to sand molding except that plaster is substituted for sand. Plaster compound is actually composed of 70-80% gypsum and 20-30% strengthener and water. Generally, the form takes less than a week to prepare, after which a production rate of 1-10 units/hr-mold is achieved with a capability to pour items as massive as 45 kg and as small as 30 g with very high surface resolution and fine tolerances.
Once used and cracked away, normal plaster cannot easily be recast. Plaster casting is normally used for nonferrous metals such as aluminium-, zinc-, or copper-based alloys. It cannot be used to cast ferrous material because sulfur in gypsum slowly reacts with iron. Prior to mold preparation the pattern is sprayed with a thin film of parting compound to prevent the mold from sticking to the pattern. The unit is shaken so plaster fills the small cavities around the pattern. The form is removed after the plaster sets.
Plaster casting represents a step up in sophistication and required skill. The automatic functions easily are handed over to robots, yet the higher-precision pattern designs required demand even higher levels of direct human assistance.
Plaster casting (of plaster or concrete)
Plaster itself may be cast – either using single use waste molds, multiple use piece molds, or molds made of flexible material such as latex rubber (which is in turn supported by an exterior mold). The finished product is, unlike marble, relatively unattractive, lacking in transparency, and so is usually painted, often in ways that give the appearance of metal. Alternatively, the first layers cast may contain colored sand so as to give an appearance of stone. By casting concrete, rather than plaster, it is possible to create sculptures, fountains, or seating for outdoor use.
Shell molding
Shell molding is also similar to sand molding except that a mixture of sand and 3-6% resin holds the grains together. Set-up and production of shell mold patterns takes weeks, after which an output of 5-50 pieces/hr-mold is attainable. Aluminium and magnesium products average about 13.5 kg as a normal limit, but it is possible to cast items in the 45-90 kg range. Shell mold walling varies from 3-10 mm thick, depending on the forming time of the resin.
There are a dozen different stages in shell mold processing that include:
1. initially preparing a metal-matched plate
2. mixing resin and sand
3. heating pattern, usually to between 505-550 K
4. inverting the pattern (the sand is at one end of a box and the pattern at the other, and the box is inverted for a time determined by the desired thickness of the mill)
5. curing shell and baking it
6. removing investment
7. inserting cores
8. repeating for other half
9. assembling mold
10. pouring mold
11. removing casting
12. cleaning and trimming.
The sand-resin mix can be recycled by burning off the resin at high temperatures.
Investment casting (lost-wax process) yields a finely detailed and accurate product, with excellent metallurgical properties.
Polystrene foam is also used in investment casting – see Lost-foam casting
After a variable lead time, usually weeks, 1-1000 pieces/hr-mold can be produced in the mass range 2.3-2.7 kg. Items up to 45 kg and as light as 30 g are possible for unit production.
The process starts by creating an injection die to the desired specifications. This die will be used to inject wax to create the patterns needed for investment casting. The patterns are attached to a central wax sprue, creating an assembly, or mold. The sprue contains the fill cup where the molten metal will be poured into the assembly.
The wax assembly is now dipped multiple times in a ceramic slurry, depending on the shell thickness desired. A layer of fine sand (usually Zircon) is added on top of each ceramic layer. This process will be repeated until the desired shell is created.
After the shell is created to the specifications desired, the wax must be removed,this is normally achieved using an autoclave. This is where the name "lost-wax process" comes from. This leaves an impression of the desired castings, which will be filled with metal. Before being cast, however, the shells must be heated up in a furnace so they don't break during the casting process.
Next, the desired metal is poured into the hot ceramic shell. The metal fills each part on the assembly, and the central sprue cavity and fill cup. The individual parts will be removed after the mold cools and the shell is removed. The shell is generally removed with water-blasting, although alternate methods can be used. What remains is the casted metal parts, but they are still attached to the sprue assembly. The individual parts are removed by cold-break (dipping in liquid nitrogen and breaking the parts off with hammer and chisel) or with large cutoff saws. Now all that remains is finishing.
First the gate, or the place where the part was connected to the sprue, must be removed. The gate is ground off to part specifications. Parts are also inspected to make sure they were cast properly, and if not are either fixed or scrapped. Depending on the investment casting facility and specifications, more finishing work can be done on-site, sub-contracted, or not done at all.
Investment casting yields exceedingly fine quality products made of all types of metals. It has special applications in fabricating very high-temperature metals, especially those which cannot be cast in metal or plaster molds and those which are difficult to machine or work.
Nonexpendable mold casting
Nonexpendable mold casting differs from expendable processes in that the mold need not be reformed after each production cycle. This technique includes at least four different methods: permanent, die, centrifugal, and continuous casting.
Permanent Mold Casting
Permanent mold casting (typically for non-ferrous metals) requires a set-up time on the order of weeks to prepare a steel tool, after which production rates of 5-50 pieces/hr-mold are achieved with an upper mass limit of 9 kg per iron alloy item (cf., up to 135 kg for many nonferrous metal parts) and a lower limit of about 0.1 kg. Steel cavities are coated with refractory wash of acetylene soot before processing to allow easy removal of the workpiece and promote longer tool life. Permanent molds have a life which varies depending on maintenance of after which they require refinishing or replacement. Casted parts from a permanent mold generally show 20% increase in tensile strength and 30% increase in elongation as compared to the products of sand casting.
The only necessary input is the coating applied regularly. Typically, permanent mold casting is used in forming iron-, aluminium-, magnesium-, and copper-based alloys. The process is highly automated.
In die-casting molten metal is injected into a mould at high pressures. Set-up time for dies is 1-2 hours, after which production rates of 20–200 pieces per hour-mould are normally obtained. Maximum mass limits for magnesium, zinc, and aluminium parts are roughly 4.5 kg, 18 kg, and 45 kg, respectively (though larger machines do exist); the lower limit in all cases is about 30 g. Die injection machines are generally large (up to 3 × 8 m) and operate at high pressures — 100 megapascals (1000 kgf/cm2) and higher, although aluminium usually is processed at lower pressure. A well-designed unit produces over 500,000 castings during the production lifetime of a single mould. While the dies used in the process are quite expensive, if a very large number of castings can be produced, significant cost savings can be achieved when a component is manufactured by die casting. The major production step is die construction, usually a steel alloy requiring a great deal of skill and fine tooling to prepare. Mostly non-ferrous materials are die-cast, such as aluminium, zinc, magnesium, and copper-based alloys.
This is the process used in the production of certain toys, notably that of model automobiles, see: die-cast toy, Matchbox
Centrifugal casting is both gravity- and pressure-independent since it creates its own force feed using a temporary sand mold held in a spinning chamber at up to 90 g (900 m/s²). Lead time varies with the application. Semi- and true-centrifugal processing permit 30-50 pieces/hr-mold to be produced, with a practical limit for batch processing of approximately 9000 kg total mass with a typical per-item limit of 2.3-4.5 kg.
Industrially, the centrifugal casting of railway wheels was an early application of the method developed by German industrial company Krupp and this capability enabled the rapid growth of the enterprise.
Small art pieces such as jewelry are often cast by this method using the lost wax process, as the forces enable the rather viscous liquid metals to flow through very small passages and into fine details such as leaves and petals. This effect is similar to the benefits from vacuum casting, also applied to jewelry casting.
Continuous Casting
Continuous casting is a refinement of the casting process for the continuous, high-volume production of metal sections with a constant cross-section. Molten metal is poured into an open-ended, water-cooled copper mould, which allows a 'skin' of solid metal to form over the still-liquid centre. The strand, as it is now called, is withdrawn from the mould and passed into a chamber of rollers and water sprays; the rollers support the thin skin of the strand while the sprays remove heat from the strand, gradually solidifying the strand from the outside in. After solidification, predetermined lengths of the strand are cut off by either mechanical shears or travelling oxyacetylene torches and transferred to further forming processes, or to a stockpile. Cast sizes can range from strip (a few millimetres thick by about five metres wide) to billets (90 to 160 mm square) to slabs (1.25 m wide by 230 mm thick). Sometimes, the strand may undergo an initial hot rolling process before being cut.
Continuous casting provides better quality product as it allows finer control over the casting process, along with the obvious advantages inherent in a continuous forming process. Metals such as steel, copper and aluminium are continuously cast, with the largest tonnage poured being steel.
Cooling rate
The rate at which a casting cools affects its microstructure, quality, and properties.
The products of sand casting and slurry-mold processes, often large with thick walls, generally cool slowly. This increases the metal's grain size, creating a coarse microstructure that lowers the strength of the casting. Coarse grains can allow elements of an alloy to separate, which also weakens the casting. But slower cooling keeps the casting metal liquid longer, which allows more gases and waste metal to escape, reducing the voids and inclusions that can weaken a casting.
Conversely, the products of die casting and metal-mold processes generally cool more quickly, resulting in a fine microstructure with small grain and less alloy segregation but more trapped gases and inclusions. On the other hand, according to the Pillings Bedworth Ratio, strength of a material is inversely proportional to the square root of its grain size. Provided de-gassing techniques are used during molten metal preparation, die cast products may have superior strength when compared with equivalent sand castings.
Like nearly all materials, metal is less dense as a liquid than a solid, and so a casting shrinks as it cools — mostly as it solidifies, but also as the temperature of the solid material drops. Compensation for this natural phenomena must be considered in two ways.
Volumetric Shrinkage
The shrinkage caused by solidification can leave cavities in a casting, weakening it. Risers provide additional material to the casting as it solidifies. The riser (sometimes called a "feeder") is designed to solidify later than the part of the casting to which it is attached. Thus the liquid metal in the riser will flow into the solidifying casting and feed it until the casting is completely solid. In the riser itself there will be a cavity showing the metal which was fed. Risers add cost because some of their material must be removed, by cutting away from the casting which will be shipped to the customer. They are often necessary to produce parts which are free of internal shrinkage voids.
Sometimes, to promote directional shrinkage, chills must be used in the mold. A chill is any material which will conduct heat away from the casting more rapidly that the material used for molding. Thus if silica sand is used for molding, a chill may be made of copper, iron, aluminum, graphite, zircon sand, chromite or any other material with the ability to remove heat faster locally from the casting. All castings solidify with progressive solidification but in some designs a chill is used to control the rate and sequence of solidification of the casting.
Linear Shrinkage
Shrinkage after solidification can be dealt with by using an oversized pattern designed for the relevant alloy. Pattern makers use special "shrink rulers" to make the patterns used by the foundry to make castings to the design size required. These rulers are 2 – 6 % oversize, depending on the material to be cast. Using such a ruler during pattern making will ensure an oversize pattern. Thus, the mold is larger also, and when the molten metal solidifies it will shrink and the casting will be the the size required by the design.
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text:
\section{Introduction}
     \label{sec1}

In recent decades, space observatories have made it possible
to study the development of solar flares in all the ranges of the electromagnetic radiation \citep{2017LRSP...14....2B}.
The brightness of flare coronal loops in the ultraviolet range is
one of the most spectacular manifestations of a solar flare
which has been observed in detail.
The complex structure of the distribution of bright loops in space indicates
the heterogeneity of the primary energy release in a flare
\citep{2003ApJ...595L.103K, 2015SoPh..290.2909R}.
Nevertheless, quasiperiodicity in the spatial distribution of bright loops in a flare arcade can often be noticed.
The Bastille day flare is a telling example of a well-observed flare arcade
extending over the photospheric neutral line \citep{2000ApJ...540.1126A, 2002ApJ...579..863S}.

According to current understanding, a thin current layer is formed
over the arcade of magnetic loops before the flare
\citep{2002A&ARv..10..313P, 2006ASSL..341.....S, 2019LRSP...16....3T}.
This current layer separates colliding magnetic fluxes
preventing them to reconnect.
This leads to the accumulation of free energy
in a non-potential magnetic field associated with the current.
Free energy is released in the form of a solar flare
during fast magnetic reconnection
when the preflare current layer is destroyed
\citep{1998A&A...331.1078O, 2000A&A...354..703S, 2007ApJ...671.2139U}.
The aim of this work is to search for a mechanism
that can lead to the destruction of the current layer that is quasiperiodic in space.

The effect of the decay of the current layer into individual
current filaments is known as tearing instability \citep{1963PhFl....6..459F, 1993SSRv...65..253S}.
This process separates the current layer along streamlines
facilitating the transition from slow reconnection to fast one.
However, it does not allow to see in which places along the current direction
one should expect an increased energy release.
The current layer decays entirely in the classical tearing instability.
From the mathematical point of view, this is due to the
absence of a wave-type solution in the direction along the current.
Often, a similar solution was sought in the interaction of the current layer
with magnetohydrodynamic (MHD) waves
\citep{1976ApJ...205..868V, 2006A&A...452..343N, 2012SoPh..277..283A}.
Also, a spatially inhomogeneous energy release was considered
as a result of the corrugation instability of a coronal arcade \citep{2017SoPh..292..184K}.
The magnetic field frozen into the plasma displaced by the instability could reconnect
with the overlying magnetic field,
leading to the heating of the unstable flux tube.

In \cite{1982SoPh...75..237S},
the heat balance inside the current layer \citep{1976SvAL....2...13S}
is considered.
In fact, a particular case
of thermal instability \citep{1965ApJ...142..531F} in the geometry of the current layer
is studied.
Investigation of the heat balance in the coronal plasma is applied
in modeling the observed properties of magnetic loops \citep{2019SoPh..294..173K, 2020PPCF...62a4016A}
and prominences \citep{2006A&A...460..573C}.
Thermal imbalance leads to the unstable growth of entropy waves \citep{2007AstL...33..309S}
affecting the stability of magnetosonic waves
\citep{2019A&A...624A..96C, 2020PhPl...27c2110P}
and causing the dispersion of slow MHD waves \citep{2019PhPl...26h2113Z}.
The heat-induced attenuation of slow waves
in the cylindrical geometry of a magnetic tube \citep{2017ApJ...849...62N}
is used to diagnose the plasma in coronal loops on the Sun \citep{2019A&A...628A.133K}.

We consider a piecewise homogeneous model of a current layer,
which consists of a magnetically neutral current layer
surrounded by a plasma with an external magnetic field.
In the equilibrium state, the plasma inside the current layer does not contain a magnetic field.
However, the disturbance of the external magnetic field can penetrate inward
when the screening currents are disturbed.
The situation of the appearance of a magnetic field in an MHD medium
that does not initially contain one is realized.
This situation is interesting in itself, and not only in the context of magnetic reconnection.
Therefore, we first consider the more general problem
of the heat balance of a homogeneous plasma without a magnetic field (Section~\ref{sec2}).
Then we apply the found solution to the particular geometry of the preflare current layer (Section~\ref{sec3}).
Finally, we consider this current layer in the context of a coronal plasma (Section~\ref{sec4}).
Our conclusions are given in Section~\ref{sec5}.

\section{Thermal Instability of a Homogeneous Plasma}
      \label{sec2}

In order to study the physical nature of the process of instability formation,
homogeneous plasma
in the single-fluid dissipative MHD approximation is considered.
The MHD approximation has been successfully used for coronal applications
for more than 50 years
(e.g. \citeauthor{2020ARA&A..58..441N}, \citeyear{2020ARA&A..58..441N}).
It imposes
some restrictions on the possible plasma processes under consideration.
First, these processes must be sufficiently slow
compared with the time of electron-ion collisions,
so that the Maxwell distribution of electrons and ions
with a common temperature is established in the plasma.
Plasma processes must be
also sufficiently slow with respect to
the inverse plasma conductivity to
neglect the displacement current in comparison with
conductive current in Maxwell's equations.
Second, the magnetic field must be weak enough
to use isotropic conductivity in the generalized Ohm's law.
Third, the velocities of the considered plasma motions
must be sufficiently small in comparison with the speed of light
so that the action of electric forces
as compared with magnetic ones can be neglected in the nonrelativistic limit.

The first condition satisfies our consideration
of the preflare state of the plasma in the solar corona,
when fast energy release does not yet take place,
and the separation of electron and ion temperatures is not important.
The second condition is consistent with the general idea of a solar flare
as a result of the process of magnetic reconnection at the zero point of the magnetic field.
The third condition is certainly valid in the context
of the observed preflare plasma velocities in the solar corona.
However, the effects of finite conductivity
during the formation of the preflare current layer cannot be neglected.
The subject of this study is the thermal balance
of the plasma in the preflare configuration,
and therefore it is assumed that Joule and viscous heating,
thermal conductivity, and radiative cooling in the energy equation are preserved.
Thus, the following set of dissipative MHD equations is
sufficient for our consideration
\citep{1958ForPh...6..437S, 2006ASSL..340.....S}:

\vspace{5mm}
\begin{displaymath}
    \frac{\partial n}{\partial t} + \div \, (n \vec{v})
    = 0 \, ,
\end{displaymath}
\begin{displaymath}
    \mu n \, \frac{\rm{d} \vec{v}}{{\rm d} t}
  = -  {\bf \nabla }  (2 n k_{_{ \rm B }} T)
  - \frac{1}{4\pi} \, ( \vec{B} \times  \mathrm{curl}  \vec{B} ) + \eta \, \Delta \vec{v} + \nu \,  {\bf \nabla }  \, \div \, \vec{v} \, ,
\end{displaymath}
\begin{displaymath}
    \frac{2 n k_{_{ \rm B }} }{\gamma -1} \,
    \frac{{\rm d} T}{{\rm d} t}
  - 2 k_{_{ \rm B }} T \, \frac{ {\rm d} n }{ {\rm d} t } \\
  = \frac{ c^2 }{ ( 4 \pi )^2 \sigma } \, ( { \mathrm{curl} } {\vec B} )^2 + \pder{}{r_\alpha} (\sigma_{\alpha \beta} v_\beta)
  + {\div} \, ( \kappa  {\bf \nabla }  T ) - \lambda \, (n,T) \, ,
  \end{displaymath}
\begin{displaymath}
    \frac{\partial {\vec B}}{\partial t}
  = { \mathrm{curl} } \, ( {\vec v} \times {\vec B} )
  - \frac{ c^2 }{ 4 \pi } \, { \mathrm{curl} }
    \left( \frac{1}{\sigma} \, { \mathrm{curl} } {\vec B} \right) ,
\end{displaymath}
\begin{equation}
    {\div} {\vec B} = 0 \, .
\label{01}
\end{equation}
Here, $ \mu =1.44 \, m_H $, $ m_H $ is the mass of the hydrogen atom,
$ k_{_{ \rm B }} $ is the Boltzmann constant,
$ \gamma $ is the heat capacity ratio,
$ \kappa $ and $ \sigma $ are the thermal and electric conductivities of the plasma,
$ \lambda\,(n,T) $ is the radiative cooling function,
$ \eta $ and $ \nu $ are viscosity ratios,
and $\sigma_{\alpha \beta}$ is the viscous stress tensor.
Transfer coefficients are isotropic in the absence of an external magnetic field.
The heat capacity ratio is assumed $ \gamma = 5/3 $ for simplicity.
$T$ is the temperature, $n$ is the palsma density, $v$ is the plasma velocity, and $B$ is the magnetic field.
System (\ref{01}) will be also used to describe a piecewise homogeneous model
of the current layer in Section~\ref{sec3}.

\subsection{Increments of Instability}
  \label{sec2.1}

The solution to Equations \ref{01}
in the form of the sum of a constant homogeneous term
and a small perturbation is sought using the following Fourier
transform with subsequent linearization in $f'$
\begin{displaymath}
f ({\vec r},t) = f_{const} + f' \, {\rm exp} \, ( - i \omega t + i (\vec{k} \vec{r}) ) \, .
\end{displaymath}
Here $ f' \equiv \{{\vec{v}'}, n', T', {\vec{B}'}\} $ are perturbation amplitudes.

Let us set ${\vec{v}}_{\rm const}=0$ and ${\vec{B}}_{\rm const}=0$.
It is worth noting that both Joule and viscous heating
turn out to be of second order
in the perturbation and can be neglected in a linear phase.
Only radiative cooling and thermal conductivity affect
the thermal balance of the plasma in a linear approximation.
The first seeks to cool the plasma,
while the second redistributes heat between regions with different temperatures.
Thus, the plasma tends to cool against the background of small perturbations.
Naturally, this does not contradict the initial heat balance.
Even if radiative cooling is not compensated
by Joule or viscous heating in an unperturbed plasma,
one can consider additional constant heating
as part of the thermal function $\lambda$.
An additional constant term creates an initial heat balance
and does not affect small perturbations, since it disappears during the linearization
\citep{1992PPCF...34..411H, 2004A&A...415..705D, 2019A&A...624A..96C}.
It is also possible to consider a more general non-constant thermal function,
but such a consideration goes beyond the physical formulation of our problem
\citep{1978ApJ...220..643R, 1993ApJ...415..335I, 2020A&A...644A..33K}.
The set of linear equations will take the following form:
\begin{equation}
\omega \, n' = n \, {(\vec{k}\vec{v}')} \, ,
    \label{02}
\end{equation}
\begin{equation}
i \omega \, n \, {\vec v}'  = i {\vec k} \, \frac{2 k_{_{ \rm B }}}{ \mu } \, ( n T' + T n' )
+k^2 \, \eta \, {\vec v}'  + \omega {\vec k} \, \frac{\nu}{n}\,n ' \, ,
    \label{03}
\end{equation}
\begin{equation}
    i \omega \, \frac{ 2 n k_{_{ \rm B }} }{ \gamma - 1 }  \, T'
  - i \omega \, 2 k_{_{ \rm B }} T \, n' =
 k^2 \, \kappa \, T' +  \pder{ \lambda }{ T } \, T' + \pder{ \lambda }{ n } \, n'  \, ,
    \label{04}
\end{equation}
\begin{equation}
i \omega \, {\vec B}' = \frac{ c^2 }{ 4 \pi \sigma } \, (k^2 \, {\vec B}'-{\vec k} \,({\vec{k}\vec{B}'})) \, ,
    \label{05}
\end{equation}
\begin{equation}
({\vec{k}\vec{B}'})=0 \, .
    \label{06}
\end{equation}

Equations \ref{02}\,--\,\ref{06} split into two subsystems of equations.
The perturbations of velocity, concentration, and temperature
enter only in the first three equations,
while the perturbation of the magnetic field enters only in Equations \ref{05} and \ref{06}.
In this regard, it is worth paying attention to a couple of nuances.

First, we assume that $\vec{B}' \ne 0$.
In this article, we will not investigate the reasons for the occurrence
of a nonzero magnetic field perturbation in an initially magnetically neutral plasma.
Such a study goes beyond the framework of our MHD approach and
requires the use of kinetic theory, such as the Weibel instability \citep{1959PhRvL...2...83W}.
In this section, we want to show that the formation of a magnetic field perturbation
creates a dedicated frequency in a broadband disturbance subject to thermal instability.
In what follows, when considering the piecewise homogeneous model of the preflare current layer (Section~\ref{sec3}),
we will assume that the perturbation of the magnetic field penetrates
into the magnetically neutral current layer from the surrounding plasma
upon dissipation of the screening currents flowing over the surface of the current layer in an unperturbed state.
The formulation of the problem implies the appearance of a magnetic field
in a medium that initially does not contain one,
and this is exactly what we expect in the region of magnetic reconnection.
External magnetic fields compensate each other inside the current layer in equilibrium state,
but they can penetrate inside when electric currents are disturbed.

Second, the division of the Equations \ref{01} into subsystems of equations does not indicate
the formation of several perturbation modes,
as is the case in a complete MHD system with a magnetic field
during the formation of entropy, Alfv\'en, and slow and fast magnetoacoustic waves.
Mode separation occurs when one dispersion relation allows several different solutions,
but here we have several dispersion relations for one solution.
It is also not a resonance between different solutions,
because we are initially looking for one solution that satisfies two conditions.
The system of Equations \ref{02}\,--\,\ref{04} describes the linear evolution of entropy and sound modes
in a nonideal hydrodynamic medium, but Equations \ref{05} and \ref{06}
additionally require the occurrence of a magnetic field perturbation.
This disturbance should not be confused with standard fast and slow magnetoacosutic waves
for which the existence of the initial guiding field is essential.
The system of Equations \ref{02}\,--\,\ref{06} describes a hydrodynamic disturbance that allows a magnetic field to arise.
This type of perturbations requires specific conditions
that occur in coronal plasma structures such as current layers only.

Two different subsystems allow us to directly determine the frequency of perturbations
which may become unstable according to the scenario described above.
Let us substitute Equation \ref{06} in Equation \ref{05}
and express the value of $k^2$.
Then we multiply Equation \ref{03} per the wave vector $\vec k$
and replace $(\vec{k}\vec{v}')$ and $k^2$
by Equations \ref{02} and \ref{05}, respectively.
Finally, we exclude one of the perturbations $n'$ and $T'$ from Equation \ref{03}
with the help of Equation \ref{04}.
Then the second perturbation is absent in the resulting equation.
Finally we get
\begin{eqnarray}
{\it \Gamma}^{\,3} \,
  &-& \left[ \,
    \frac{ 1 }{ \tau_\sigma - \tau_\eta }
    \left( 1 + \left({ \frac{1}{\gamma-1} - \frac{\tau_\kappa}{\tau_\sigma} }\right)^{-1} \right) \,+\,
     \frac{ - \alpha }{ \tau_\lambda } \left({ \frac{1}{\gamma-1} - \frac{\tau_\kappa}{\tau_\sigma} }\right)^{-1}
    \right]
    {\it \Gamma}^{\,2}   \nonumber  \\
    &+&\left[ \,\frac{ 1 }{ \tau_\sigma - \tau_\eta } \,\, \frac{ \beta - \alpha }{ \tau_\lambda }
   \left({ \frac{1}{\gamma-1} - \frac{\tau_\kappa}{\tau_\sigma} }\right)^{-1}
   \right] {\it \Gamma} = 0 \, .
    \label{07}
\end{eqnarray}
Here the growth rate of the instability is $ {\it \Gamma} = - i \omega $.
Positive values of $ {\it \Gamma} $ correspond to the exponential growth of the perturbation in time
while negative values indicate stabilization of the initial perturbation.
Also we introduce the new notations for the logarithmic derivatives of the cooling function
\begin{equation}
    \alpha
  = \pder{ {\rm \, ln} \, \lambda }{ {\rm \, ln} \, T } \, ,
  \qquad
  \beta
  = \pder{ {\rm \, ln} \, \lambda }{ {\rm \, ln} \, n } \, ,
    \label{08}
\end{equation}
and the characteristic times
\begin{equation}
    \tau_\sigma
  =  \frac{ \mu \, \nu_m}{2k_{_{ \rm B }} T} \, ,
  \qquad
  \tau_\eta
  = \frac{ \eta + \nu }{ 2k_{_{ \rm B }} T n } \, ,
\qquad
\tau_\kappa
  = \frac{ \mu \, \kappa }{ (2k_{_{ \rm B }})^2 T n } \, ,
\qquad
    \tau_\lambda
  =  \frac{ 2 k_{_{ \rm B }} T n }{ \lambda } \, ,
    \label{09}
\end{equation}
of the magnetic resistivity, viscosity, thermal conduction, and optically thin radiation, respectively.
Derivatives of the heating function do not affect the development of the instability,
as it is taken constant in time in this work.
The magnetic viscosity is denoted as
\begin{displaymath}
    \nu_m
  =  \frac{ c^2 }{4 \pi \sigma} \, .
\end{displaymath}
We mention that the Equations for characteristic times \ref{09}
are written in such a manner to provide the clarity of the final result.
For this reason, they do not coincide,
for example, with similar equations in \citet{1982SoPh...75..237S}.

We also introduce the dimensionless parameter
\begin{equation}
\delta =
         \left({ \frac{1}{\gamma-1} - \frac{\tau_\kappa}{\tau_\sigma} }\right)^{-1} \, ,
    \label{10}
\end{equation}
and effective viscous time
\begin{equation}
\tau_\nu =
         { \tau_\sigma - \tau_\eta } \, .
    \label{11}
\end{equation}
Then, Equation \ref{07} can be written in the following simple form
\begin{equation}
{\it \Gamma}^{\,3} \,
  - \left[ \,
\frac{ 1 + \delta }
         { \tau_\nu } \,+\,
     \frac{ - \alpha \, \delta }{ \tau_\lambda } \,
        \right]
    {\it \Gamma}^{\,2}
    + \left[ \, \frac{ 1 }{ \tau_\nu } \,\,
       \frac{ (\beta - \alpha) \, \delta }{ \tau_\lambda } \,\right] {\it \Gamma} = 0 \, .
    \label{12}
\end{equation}
\subsection{Features of the Instability}
\label{sec2.2}

\begin{figure}
\begin{center}
\includegraphics*[width=\linewidth]{fig1.eps}
\end{center}
\caption{(a) Radiative loss function of an optically thin medium
$L(T)$ based on the CHIANTI atomic database \citep{2019ApJS..241...22D}
for the coronal abundance of elements \citep{2012ApJ...755...33S} and $n=10^8 {\rm cm}^{-3}.$
(b) The logarithmic derivative of the radiative cooling function $\alpha$
with respect to the logarithm of temperature for the same conditions.}
\label{fig1}
\end{figure}
In the current section, Equation \ref{12} is applied to the physics of solar flares.
To this end, the characteristic values of the quiet coronal plasma are used as a starting point:
$n=10^8 {\rm \, cm}^{-3}$, $T=10^6 {\rm \, K}$.
The same instability will be considered
in broad intervals of
plasma densities and temperatures
in Section~\ref{sec4}.
Anomalous conductivity $\sigma=10^{12} {\rm \, s}^{-1}$ caused mainly by the ion-acoustic turbulence
is usually applied in the context of the emerging preflare current layer \citep{2006ASSL..341.....S}.
Viscosity changes the effective viscous time according to Equation \ref{11}.
The coefficient of dynamic viscosity is estimated as \citep{1986ApJ...306..730H}
\begin{displaymath}
    \eta
    \approx 10^{-16} \, T^{\, 5/2} \, .
\end{displaymath}
Hereinafter, all quantities are measured in Gaussian units in practical equations.
Using Equations \ref{09} we are convinced that $\tau_\eta \ll \tau_\sigma$
here and for all further calculations in the article.
Therefore we set $\eta=0$, $\nu=0$ in what follows.
We also use a common representation of the radiative cooling function $\lambda \,(n,T)=n^2L(T)$,
where $L(T)$ is
the radiative loss function
of an optically thin medium.
Figure \ref{fig1}a shows the function $L(T)$ based on the
CHIANTI version 9
atomic database \citep{2019ApJS..241...22D}
for coronal abundance elements
(see file ${\rm sun\_coronal\_2012\_schmelz\_ext.abund}$ in the standard CHIANTI distribution and
\citeauthor{2012ApJ...755...33S}, \citeyear{2012ApJ...755...33S}).
The temperature dependence of the coefficient $\alpha$ is shown in Figure \ref{fig1}b.
The plasma thermal conductivity is considered as a free parameter in this section.

The roots of Equation \ref{12}
depend on the dimensionless parameter $\delta$.
The characteristic time $\tau_\kappa$ is directly proportional to the coefficient of thermal conductivity $\kappa$,
while the characteristic time $\tau_\sigma$ is inversely proportional to the electrical conductivity $\sigma$
according to the definitions in Equations \ref{09}.
Therefore the fraction $\tau_\kappa / \tau_\sigma$ is proportional to
both thermal and electrical conductivities of the plasma in Equation \ref{10}.
In addition, both of them have similar physical nature
associated with the mean free path of the particles.
It is expected that both of them increase or decrease
under similar conditions in the plasma.
For simplicity, in this section we treat electrical conductivity as a constant
and vary thermal conductivity.
Figure \ref{fig2}a shows the dependence of the parameter $\delta$
on the thermal conductivity measured in units
of the Spitzer's thermal conductivity \citep{1953PhRv...89..977S}
\begin{displaymath}
    \kappa_{\, \rm e}
    \approx 9 \times 10^{-7} \, T^{5/2} \, .
\end{displaymath}

\begin{figure}
\begin{center}
\includegraphics*[width=\linewidth]{fig2.eps}
\end{center}
\caption{
Profiles depending on the thermal conductivity of the plasma:
(a) parameter $\delta$ (Equation \ref{10}), (b) discriminant $D$,
(c) root ${\it \Gamma}_1$, and (d) root ${\it \Gamma}_2$ of Equation \ref{12}.
Thermal conductivity is measured in units of the
classical electronic thermal conductivity
calculated for $T=10^6$ K
\citep{1953PhRv...89..977S}.}
\label{fig2}
\end{figure}
As one can see, $|\,\delta\,|<1$ for all $\kappa$,
except an interval $2 \times 10^{-8} \,\kappa_{\, \rm e} < \kappa < 6 \times 10^{-8} \,\kappa_{\, \rm e}$.
The sign of the parameter $\delta$ changes
when the plasma thermal conductivity decreases
to $\kappa \lesssim 3 \times 10^{-8} \,\kappa_{\, \rm e}$.
For example, if thermal conductivity is suppressed
by a perturbation of the magnetic field,
then ionic thermal conduction becomes more efficient \citep{1958PhRv..109....1R}
\begin{displaymath}
    \kappa_{\, \rm i}
    \approx 2 \times 10^{-17} \,
    \frac{ n^2 }{ T^{1/2} B'^2 } \, .
\end{displaymath}
So in Section \ref{sec3},
the thermal conductivity inside the current layer is suppressed
by a perturbation of the magnetic field
directed along the external magnetic field
(for more details on the field configuration see Section \ref{sec3}).
The magnitude of the required perturbation
of the magnetic field can be found from the evaluation
$\kappa_{\, \rm i} \approx 3 \times 10^{-8} \,\kappa_{\, \rm e}$.
The amplitude of the magnetic field perturbation $B'~\gtrsim~0.01~{\rm \, G}$
is sufficient to change the sign of the parameter $\delta$.
This will be important for further discussion.

The roots of Equation \ref{12} are as follows:
\begin{displaymath}
    {\it \Gamma}_0 = 0 \, ,
\end{displaymath}
\begin{displaymath}
    {\it \Gamma}_{1,2} = \frac{1}{2}\left\{
    \left( \,
\frac{ 1 + \delta }
         { \tau_\nu } \,+\,
     \frac{ - \alpha \, \delta }{ \tau_\lambda } \,
        \right) \pm \left[ \left( \,
\frac{ 1 + \delta }
         { \tau_\nu } \,+\,
     \frac{ - \alpha \, \delta }{ \tau_\lambda } \,
        \right)^2 - 4 \, \frac{ 1 }{ \tau_\nu } \, \frac{ (\beta - \alpha) \, \delta }{ \tau_\lambda }  \right]^{1/2}\right\}
\end{displaymath}
The root ${\it \Gamma}_0$ is not of interest here,
since it corresponds to the transition to a new stationary state,
which differs from the initial one by the magnitude of the perturbation.
The relation $\tau_\nu / \tau_\lambda$ is much smaller than 1 for the described conditions of the solar corona.
Therefore, the roots ${\it \Gamma}_{1,2}$ can be expanded in small parameter $\tau_\nu / \tau_\lambda $.
Keeping only zero-order terms, one obtains:
\begin{equation}
    {\it \Gamma}_1 \simeq
    \frac{ 1+\delta }{ \tau_\nu } \, ,
    \qquad
        {\it \Gamma}_2 \simeq
    \frac{ \beta-\alpha }{ \tau_\lambda } \, \frac{ \delta }{ 1+\delta } \, .
    \label{13}
\end{equation}

Figure \ref{fig2}b shows the dependence of the discriminant $D$
in Equation \ref{12} on the thermal conductivity, while
Figure \ref{fig2} c and d shows the roots ${\it \Gamma}_{1,2}$.
The exact calculation of the roots ${\it \Gamma}_{1,2}$ completely coincides
with the approximate Equations \ref{13} in the scale of Figure \ref{fig2} c and d.
Differences are observed only
in the region of rapid growth of $|\,\delta\,|$,
where the discriminant $D$ also tends to infinity,
and in the region where the discriminant $D$ is negative (Figure \ref{fig2}b).
The root ${\it \Gamma}_{1}$ has a discontinuity in the first region (Figure \ref{fig2}c),
while the root ${\it \Gamma}_{2}$ has a discontinuity in the second region (Figure \ref{fig2}d).
In these areas, the linear approximation of the problem of small perturbations is unsuitable.
Changing the initial parameters $n$, $T$, and $\sigma$
within the limits which are acceptable for the conditions of the solar corona
stretches or compresses Figure \ref{fig2} along the coordinate axes,
but does not make any qualitative changes in these plots.

The figure shows that ${\it \Gamma}_1 \gg {\it \Gamma}_2$ for almost all values $\kappa$
except for a narrow interval near $\kappa = 4 \times 10^{-8} \,\kappa_{\, \rm e}$
where ${\it \Gamma}_1$ is negative and ${\it \Gamma}_2$ is positive.
This means that the instability described by the root ${\it \Gamma}_1$ should grow
much faster than the instability described by the root ${\it \Gamma}_2$
everywhere except in this narrow interval.
In the geometry of the preflare current layer in the solar corona,
the spatial scale of the root ${\it \Gamma}_1$ does not satisfy the MHD approximation used (Section \ref{sec4.1})
and we should use a higher frequency approximation to study it further.
Therefore, in what follows, we will focus on the root ${\it \Gamma}_2$ and assume that
the value of the thermal conductivity satisfies the condition ${\it \Gamma}_2 > {\it \Gamma}_1$.

\section{Current Layer Model}
\label{sec3}

We consider the piecewise homogeneous model
of the preflare current layer, presented by \cite{1982SoPh...75..237S}.
The current layer is located in the $(x,z)$ plane (Figure \ref{fig3}).
The $z$-axis complements the right triplet $(x,y,z)$ and is directed toward the reader in Figure \ref{fig3}.
The plasma concentration and temperature inside the layer
are equal to $n_s$ and $T_s$, respectively.
The current layer is assumed magnetically neutral, $B_s = 0$,
without any directed plasma flows, i.e. $v_s = 0$.
The half-thickness of the current layer $a$
is much smaller than its half-width $b$.
When considering the preflare non-reconnecting current layer,
$b \to \infty$ is assumed.
As a consequence, $\partial / \partial {x}=0$ in such model.
This means that
we neglect the evolution of the current layer
along the $x$-axis, such as the tearing instability \citep{1963PhFl....6..459F}.
We focus on the structure of the current layer along the $z$-axis.
The inner region of the current layer is separated from the outer plasma
by a tangential discontinuity \citep{2015AdSpR..56.2779L}.
Outside the layer, we denote the concentration and the temperature of the homogeneous plasma
as $n_0$ and $T_0$, correspondingly.
A uniform magnetic field $B_0$ is directed against the $x$-axis for positive $y$
and along the $x$-axis for negative $y$.
Thus, the current in the layer is directed along the $z$-axis.
In order to study the effect of thermal balance on the structural stability
of a preflare current layer, the effects of viscosity, electrical and thermal conductivity,
and radiative cooling are considered inside the current layer,
but these effects are insignificant outside.
An important difference between the model considered here
and the \cite{1982SoPh...75..237S} model is the possibility
of penetration of a magnetic field perturbation inside the current layer.
Mathematically, this comes to considering the current layer interior
in the magnetohydrodynamic approximation rather than in the hydrodynamic one.
\begin{figure}
\begin{center}
\includegraphics*[width=0.75\linewidth]{fig3.eps}
\end{center}
\caption{Location of the current layer in the coordinate system.}
\label{fig3}
\end{figure}

\subsection{Outside the Current Layer}
\label{sec3.1}

Following \cite{1982SoPh...75..237S},
we set $\sigma \to \infty$, $\kappa=0$, $\lambda=0$, $\eta=0$, and $\nu=0$
in the set of Equations \ref{01} outside the current layer.
Plasma density contrast inside and outside the super-hot turbulent-current layers is about 5
(see Section 8.5.3 in \citeauthor{2006ASSL..341.....S}, \citeyear{2006ASSL..341.....S}).
Kinetic models give the same values \citep{2015PhPl...22k2902K, 2017A&A...600A..78P}.
Wherein, the external plasma could radiate up to a factor of 100 less efficiently than the internal one.
In other words, the characteristic timescales of radiative processes outside the layer and those inside it
(including the characteristic timescales of the perturbation and of the other non-adiabatic processes)
could differ by two orders of magnitude, allowing one to neglect the effects of radiation in the external plasma.
Moreover, we suppose that the considered preflare current layer is more similar
to the neutral current layer by Syrovatskii,
in which the density contrast can be much higher \citep{1976SvAL....2...13S}.
In addition, the plasma is assumed to be at rest, i.e. $v_0=0$.
The solution is sought in the form of a periodic perturbation
along the $z$-axis in Figure \ref{fig3} which decays exponentially with distance from the current layer:
\begin{displaymath}
    f(y,z,t)
  = f_0
  + f_1(y) \, {\rm exp} \, ( - i \omega t + i k_z z ) \, ,
\end{displaymath}
\begin{displaymath}
    f_1(y) _{\rm top}=f_{\rm 1 \, top} \, {\rm exp} \, [ - k_{y1} (y-a) ] \, ,
    \qquad
    f_1(y) _{\rm bottom}=f_{\rm 1 \, bottom} \, {\rm exp} \, [ k_{y1} (y+a) ] \, ,
\end{displaymath}
with perturbation amplitudes
\begin{displaymath}
    f_{\rm 1 \, top} \equiv \{{v_{y1}, v_{z1}, n_1, T_1, B_{x1}}\} \, ,
    \qquad
    f_{\rm 1 \, bottom} \equiv \{{-v_{y1}, v_{z1}, n_1, T_1, -B_{x1}}\} \, ,
\end{displaymath}
on either side outside the current layer, respectively.
Here, $\omega$ is the perturbation frequency,
$k_z$ and $k_{y1}$ are the perturbation wave numbers along the $z$ and $y$ axes, respectively,
and $a$ is the half-thickness of the current layer.
Index ``$1$'' refers to quantities outside the layer.
Thus, we are looking for a solution in the form of a perturbation
that propagates
through the surface of the current layer
and decays with distance from it.

Based on the symmetry of the problem,
the set of Equations \ref{01} is considered only for the upper half space.
Neglecting the squares of the perturbed quantities,
one finds the linearized system of equations:
\begin{equation}
    i \omega \, n_1
  = - k_{y1} \, n_0 v_{y1} + i k_z \, n_0 v_{z1} \, ,
    \label{14}
\end{equation}
\begin{equation}
    i \omega \, \mu n_0 v_{y1}
  = - k_{y1} \, 2k_{_{ \rm B }}  (n_0 T_1 + T_0 n_1)
    + k_{y1} \, \frac{ B_0  }{ 4 \pi }\,B_{x1}  \, ,
    \label{15}
\end{equation}
\begin{equation}
    i \omega \, \mu n_0 v_{z1}
  = i k_{z} \, 2k_{_{ \rm B }}  (n_0 T_1 + T_0 n_1)
  - i k_{z} \, \frac{ B_0  }{ 4 \pi } \, B_{x1} \, ,
    \label{16}
\end{equation}
\begin{equation}
    ( \gamma - 1 ) \, T_0 n_1 = n_0 T_1 \, ,
    \label{17}
\end{equation}
\begin{equation}
    i \omega \, B_{x1}
  = k_{y1} \, B_0 v_{y1} - i k_z \, B_0 \, v_{z1} \, .
    \label{18}
\end{equation}
The dispersion relation for perturbations outside the current layer is determined by
equating the determinants of a homogeneous system of linear Equations \ref{14}\,--\,\ref{18} to zero
\begin{equation}
    k_{y1}^2 = k_z^2 - \frac{\omega^2}{ V_S^2 + V_A^2 } \, ,
    \label{19}
\end{equation}
where the sound and the Alfv\'en speeds are denoted as
\begin{equation}
    V_S = \sqrt{ \frac{ 2 \gamma k_{_{ \rm B }} T_0 }{ \mu }} \, ,
    \qquad
        V_A = \frac{B_0}{ \sqrt{ 4 \pi n_0 \mu } } \, ,
    \label{20}
\end{equation}
respectively.
The dispersion relation in Equation \ref{19} describes a fast magnetoacoustic wave
propagating over the surface of the current layer perpendicular to the magnetic field.

\vspace{1mm}

\subsection{Inside the Current Layer}
\label{sec3.2}

Dissipative effects of Joule and viscous heating,
thermal conductivity, and radiative cooling
should be considered inside the current layer.
The current layer is assumed magnetically neutral, $B_s=0$,
without any directed plasma flows, i.e. $v_s=0$.
The solution is sought in the same form as outside the current layer
\begin{displaymath}
    f(y,z,t) = f_s
             + f_2(y) \, {\rm exp} \, (-i\omega t + ik_zz) \, .
\end{displaymath}
The perturbations decrease exponentially along the $y$-axis
when moving from the upper boundary of the current layer
\begin{displaymath}
    f_2(y)_{\rm top} =f_{\rm 2 \, top} \, {\rm exp} \, [ - k_{y2} (a - y) ] \, ,
\end{displaymath}
\begin{displaymath}
    f_{\rm 2 \, top} \equiv \frac{1}{2} \, {\rm exp} (k_{y2} a) \, \{ v_{y2}, v_{z2}, n_2, T_2, B_{x2} \} \,
\end{displaymath}
and when moving from the lower boundary
\begin{displaymath}
    f_2(y)_{\rm bottom} =f_{\rm 2 \, bottom} \, {\rm exp} \, [ - k_{y2} (a + y) ] \, ,
\end{displaymath}
\begin{displaymath}
    f_{\rm 2 \, bottom} \equiv \frac{1}{2} \, {\rm exp} (k_{y2} a) \, \{ -v_{y2}, v_{z2}, n_2, T_2, -B_{x2} \} \, .
\end{displaymath}
Here, index ``$2$'' refers to perturbations inside the layer,
$\frac{1}{2} \, {\rm exp} (k_{y2} a)$ is a scale factor that unifies the solution for all thicknesses of the current layer.
Inside the current layer, perturbations coming from the upper and lower boundaries add up
\begin{displaymath}
f_2(y) = f_2(y)_{\rm top} + f_2(y)_{\rm bottom} \, .
\end{displaymath}

The resulting dependences of the perturbations on the coordinate $ y $
are hyperbolic functions.
The sum of the perturbations which are odd in the $y$ direction gives a hyperbolic sine
\begin{displaymath}
\left\{
\begin{array}{c}
v_{y2}(y)\\
B_{x2}(y)
\end{array}
\right\} =
\left\{
\begin{array}{c}
v_{y2}\\
B_{x2}
\end{array}
\right\}
{\rm sinh} \, (k_{y2} y) \, ,
\end{displaymath}
The sum of the perturbations which are even in $y$ gives a hyperbolic cosine
\begin{displaymath}
\left\{
\begin{array}{c}
v_{z2}(y)\\
n_2(y)\\
T_2(y)
\end{array}
\right\} =
\left\{
\begin{array}{c}
v_{z2}\\
n_2\\
T_2
\end{array}
\right\}
{\rm cosh} \, (k_{y2} y) \, .
\end{displaymath}
Real values of $k_{y2}$ determine the effective thickness of the skin depth of the layer,
that is, the distance to which the disturbance of the layer boundary penetrates.
On the other hand, a standing wave is formed inside the current layer along the $y$-axis at imaginary $k_{y2}$.
The value of the wave number $k_{y2}$ is not prescribed and can be determined from the solution,
however, this is not the purpose of this article.
The solution describes the plasma motion which is symmetric about the $(x,z)$ plane.

The set of Equations \ref{01} is linearized as in the previous section:
\begin{equation}
    i \omega \, n_{2}
  = k_{y2} \, n_s v_{y2} + i k_z \, n_s v_{z2} \, ,
    \label{21}
\end{equation}
\begin{equation}
    i \omega \, \mu n_s v_{y2}
  = k_{y2} \, 2 k_{_{ \rm B }} ( n_s T_2 + T_s n_2 ) + (k_z^2-k_{y2}^2) \,
  \eta v_{y2} - i \omega k_{y2} \frac{\nu}{n_s} n_2 \, ,
    \label{22}
\end{equation}
\begin{equation}
    i \omega \, \mu n_s v_{z2}
  = i k_{z} \, 2 k_{_{ \rm B }} ( n_s T_2 + T_s n_2 ) + (k_z^2-k_{y2}^2) \,
  \eta v_{z2} - i \omega i k_{z} \frac{\nu}{ n_s} n_2 \, ,
    \label{23}
\end{equation}
\begin{equation}
    i \omega \, \frac{ 2 k_{_{ \rm B }} n_s }{ \gamma - 1} \, T_2
  - i \omega \, 2 k_{_{ \rm B }} T_s \, n_2 =  ( k_z^2 - k_{y2}^2 ) \, \kappa T_2
   + \pder{ \lambda }{ T } \, T_2 + \pder{ \lambda }{ n } \, n_2 \, ,
    \label{24}
\end{equation}
\begin{equation}
    i \omega \, B_{x2}
  = (k_z^2 - k_{y2}^2) \, \nu_m B_{x2} \, .
    \label{25}
\end{equation}
Equation \ref{25}, like the set of Equations \ref{05} and \ref{06}, can be satisfied at $B_{x2}=0$,
however, if the perturbation of the magnetic field, $B_{x2}\ne0$, penetrates into the current layer,
then Equation \ref{25} gives an additional dispersion relation independent of the $B_{x2}$ value.
After expressing the difference $k_z^2 - k_{y2}^2$ from Equation \ref{25},
we can exclude it from the remaining Equation \ref{21}\,--\,\ref{25}.
Making transformations similar to those in Section \ref{sec2.1},
gives again Equation \ref{12} with $n=n_s$ and $T=T_s$.
It is worth noting that the possibility to directly determine the instability increment
is due to the simultaneous presence of two dispersion relations in Equations \ref{21}\,--\,\ref{25} at once:
the first follows from Equations \ref{21}\,--\,\ref{24}, and the second one from Equation \ref{25}.
An important feature of Equations \ref{21}\,--\,\ref{25}
is that the wave vector $k_z$ turns out to be perpendicular
to the arising perturbation of the magnetic field $B_{x2}$.
This promotes the suppression of thermal conductivity
along the $z$-axis (in the direction of the current)
and the formation of a thermal instability (see Section \ref{sec2.2}).

\vspace{1mm}

\subsection{Boundary of the Current Layer}
\label{sec3.3}

The considered model of the current layer has no
plasma motion neither outside nor inside the layer
in equilibrium
($v_0=0, v_s=0$),
but a magnetic field jump occurs at the layer boundary.
A tangential discontinuity in MHD corresponds to such conditions \citep{2015PhyU...58..107L}.

The sum of the gas-dynamic and magnetic pressure should be equal
on the different sides of the tangential discontinuity \citep{1956TrFIAN...13..64S}.
In a linearized form, it looks as follows
\begin{equation}
    n_0 T_1 + T_0 n_1 - \frac{ B_0 B_{x1} }{ 8 \pi k_{_{ \rm B }} }
  = ( n_s T_2 + T_s n_2 ) \, {\rm cosh} \, ( k_{y2} a ) \, .
    \label{26}
\end{equation}
The left side of Equation \ref{26} can be expressed in terms of the perturbation $v_{y1}$
using Equation \ref{15}.
The right side of Equation \ref{26} can be expressed in terms of the perturbation $v_{y2}$
using Equations \ref{21}\,--\,\ref{23}
and substituting $k_z^2 - k_{y2}^2$ from Equation \ref{25}.
Then Equation \ref{26} takes the form
\begin{equation}
    - \, \frac{n_0}{n_s} \, \frac{v_{y1}}{k_{y1}}
  = \frac{\tau_\nu}{\tau_\sigma} \, \frac{v_{y2}}{k_{y2}} \, {\rm cosh} \, ( k_{y2} a ) \, .
    \label{27}
\end{equation}
Here, the Equations \ref{09} are also used with substitutions $n=n_s$ and $T=T_s$.

Velocity perturbations distort the surface of the tangential discontinuity.
For reasons of continuity, the velocity perturbation on both sides of the discontinuity
should have the same magnitude and direction
\begin{equation}
    v_{y1}^2 + v_{z1}^2 = v_{y2}^2 \, {\rm sinh}^2 \, ( k_{y2} a ) + v_{z2}^2 \, {\rm cosh}^2 \, ( k_{y2} a ) \, ,
    \label{28}
\end{equation}
\begin{equation}
    \frac{v_{z1}}{v_{y1}} = \frac{v_{z2} \, {\rm cosh} \, ( k_{y2} a )}{v_{y2} \, {\rm sinh} \, ( k_{y2} a )} \, .
    \label{29}
\end{equation}
Equation \ref{28} is then rewritten as
\begin{equation}
    v_{y1} = \pm \, v_{y2} \, {\rm sinh} \, ( k_{y2} a ) \, ,
    \label{30}
\end{equation}
where the choice of sign depends on the signs of perturbations $v_{y1}$ and $v_{z1}$.

Let us divide Equation \ref{30} by Equation \ref{27}
\begin{equation}
    \pm \, \frac{\tau_\nu}{\tau_\sigma} \, \frac{n_s}{n_0} \, k_{y1} = k_{y2} \, {\rm tanh} \, ( k_{y2} a ) \, .
    \label{31}
\end{equation}
Equation \ref{31} differs from Equation 23 of \cite{1982SoPh...75..237S}
by a coefficient $ \tau_\nu / \tau_\sigma$ determining the role of viscosity
in the formation of the structure of the preflare current layer.
Note that the right side of Equation \ref{31} is positive for any real $k_{y2}$.
This means that $ \pm(\tau_\nu / \tau_\sigma)$ should also be positive
for physically meaningful values $k_{y1}$.
Substitution of the wave numbers $k_{y1}$ and $k_{y2}$
from Equations \ref{19} and \ref{25}, respectively,
gives the dispersion relation
which relates the instability increment to the wave number $k_z$
\begin{equation}
    \left( \frac{\tau_\nu}{\tau_\sigma} \,
    \frac{n_s}{n_0}\right)^2
    \left[ k_z^2 + \frac{ {\it \Gamma}^2 }{ V_S^2 + V_A^2 } \right]
  = \left[ \, k_z^2 +\frac{ {\it \Gamma} }{ \nu_m } \,
    \right]
    {\rm tanh}^2
    \left\{ a
    \left[ \, k_z^2 + \frac{ {\it \Gamma} }{ \nu_m } \,
    \right]^{1/2}
    \right\} \, .
    \label{32}
\end{equation}
We determine the growth increment, $ {\it \Gamma} = - i \omega $, from Equation \ref{12},
which is identical to the solution of Equations \ref{21}\,--\,\ref{25}.
Then we determine the spatial period of the
instability, $ l = 2 \pi / k_z $, from Equation \ref{32}.

\section{Thermal Instability of the Current Layer}
\label{sec4}

Now the growth rate of the instability can be calculated using Equation \ref{12}
and the corresponding spatial period of the perturbation can be found from Equation \ref{32}.
For this aim, the appropriate values of the parameters
of the current layer and the surrounding plasma should be chosen.
Taking into account the possibility of plasma gathering by magnetic fields
and its heating during the formation of the current layer
before the onset of the studied instability,
the range of values $n_0=10^8-10^{12} {\rm \, cm}^{-3}$,
$n_s / n_0=10-10^3$, $T_0=10^6 {\rm \, K}$, $T_s=10^6-10^8 {\rm \, K}$,
$B_0=1-10^2 {\rm \, G}$, $\sigma=10^{11} {\rm \, s}^{-1}$, $a=10^5-10^7 {\rm \, cm}$
is considered.
This range covers all the reasonable parameters of the coronal plasma.
As one can see from the first two brackets on the left hand side of Equation \ref{32},
the effect of an increase of viscosity is the opposite to an increase in the density jump.
Therefore, no viscosity is introduced, because
its effect is taken into account in the density jump ($\eta=0$, $\nu=0$).
In addition, the viscosity effect is small ($\tau_\nu \approx \tau_\sigma$)
in the investigated range of coronal plasma parameters.

\subsection{Growth Increment in the Current Layer}
\label{sec4.1}

The instability occurs when the roots of Equation \ref{12} are positive.
The roots ${\it \Gamma}_1$ and ${\it \Gamma}_2$ are real numbers
everywhere except for a narrow interval where $D<0$ (Figure \ref{fig2}b).
In this interval, the roots become complex.
Three unstable solutions are possible:
the left branch of ${\it \Gamma}_1$ (Figure \ref{fig2}c),
the positive part of the right branch of ${\it \Gamma}_1$ (also Figure \ref{fig2}c),
and the left branch of ${\it \Gamma}_2$ (Figure \ref{fig2}d).
As one can see, $|\,{\it \Gamma}_1\,| \gg |\,{\it \Gamma}_2\,|$ everywhere
except perhaps in a small area near $D=0$ (see Figure \ref{fig2}b).
Calculation of the instability scale over the entire range
of coronal plasma parameters described above
gives $l\lesssim10^4 {\rm \, cm}$.
It is less than the corresponding Larmor radius of the proton
for most of the values of plasma parameters.
Moreover, complex values of ${\it \Gamma}_1$ for $D<0$ lead to complex values of $k_z$,
which corresponds to the spatial attenuation of the perturbation at the same scales
($l\lesssim10^4 {\rm \, cm}$).
The presence of viscosity can only increase the value ${\it \Gamma}_1$
as seen from Equations \ref{11} and \ref{13}.
Therefore, it further reduces the scale of instability.

The MHD approximation is incorrect for the description of the plasma at such scales.
Therefore, in this article we cannot say whether such instability appears in a more general kinetic description.
Remaining within the framework of the MHD, we further consider the root ${\it \Gamma}_1$ physically meaningless.
In any case, if the instability associated with the root ${\it \Gamma}_1$ exists
in the kinetic description and dominates the instability with an increment ${\it \Gamma}_2$,
there is a narrow interval of plasma thermal conductivities $\kappa$
where ${\it \Gamma}_1$ is negative and ${\it \Gamma}_2$ is positive,
and an instability occurs due to the root ${\it \Gamma}_2$ (Figure \ref{fig2}).
We assume that the thermal conductivity is suppressed
by the perturbation of the magnetic field in the current layer,
which triggers the instability.
Note that, for further reasoning, it is not important
which process led to the suppression of the thermal conductivity.
The space scale of the instability \ref{32} does not depend
on the exact value of the thermal conductivity coefficient
and can be calculated for any range of coronal plasma parameters.

The negative right branch of the root ${\it \Gamma}_2$ indicates
the stabilizing effect of the high thermal conductivity of the plasma.
However, if, for some reason,
the thermal conductivity falls below the threshold value
$\delta=-1$ (see Equation \ref{13} and Figure \ref{fig2}d),
${\it \Gamma}_2$ becomes positive and an instability occurs.
The value $\beta-\alpha$ is positive over the entire range
of the above-described coronal plasma conditions.
As it was shown in Section \ref{sec2.2}, the transverse magnetic field
can cause a decrease of the thermal conductivity.
Equations \ref{21}\,--\,\ref{25} allow the
perturbations of the $x$-component of the magnetic field
to appear inside the current layer.
This field is actually perpendicular to $\nabla T$
in the current layer under consideration.
However, the specific nature of the suppression of the thermal conductivity
is not very important for further considerations.
It is enough for us to assume that the thermal conductivity
went down for some reason ($\tau_\kappa\ll\tau_\sigma$).
Then the growth rate of the instability tends to the value
\begin{equation}
   {\it \Gamma} =
    \frac{ 2 }{ 5 } \, \frac{ \beta-\alpha }{ \tau_\lambda }
    \label{33}
\end{equation}
(see Equations \ref{13} and \ref{10}).
The growth time of the instability is proportional to the characteristic time of the
plasma cooling and depends
on the logarithmic derivatives of the radiative cooling function
(with respect to concentration and temperature).
Thermal instability criteria \citep{1965ApJ...142..531F} in our notation can be written as follows:
$$\alpha<0 \qquad \eqno({\rm isochoric}) \, ,$$
$$\alpha<\beta-1 \qquad \eqno({\rm isobaric}) \, ,$$
$$\alpha<-\frac{\beta-1}{\gamma-1} \qquad \eqno({\rm isentropic}) \, ,$$
where $\beta=2$, $\gamma=5/3$.
Thus, the criteria for isochoric, isobaric, and isentropic instabilities are
$\alpha<0$, $\alpha<1$, and $\alpha<-3/2$, respectively.
Figure \ref{fig1}b shows that the isobaric criterion of thermal instability is fulfilled
for the entire range of coronal plasma parameters.
We expect that the instability discussed in this work is a special case of
the condensation mode of the isobaric thermal instability.

\subsection{Spatial Period in the Current Layer}
\label{sec4.2}

\begin{figure}
\begin{center}
\includegraphics*[width=\linewidth]{fig4.eps}
\end{center}
\caption{The spatial period of the instability depending on the temperature of the current layer.
Parameters of the coronal plasma: $n_0=10^{10} {\rm \, cm}^{-3}$,
$n_s/n_0=10$,
$a=10^5 {\rm \, cm}$,
$B_0=100 {\rm \, G}$.
One of the parameters changes in each figure:
(a) $n_0=10^8 {\rm \, cm}^{-3}$ (circles), $n_0=10^9 {\rm \, cm}^{-3}$ (triangles),
$n_0=10^{10} {\rm \, cm}^{-3}$ (squares),
solid lines show analytical solutions (Equation \ref{35});
(b) $n_s/n_0=10$ (circles), $n_s/n_0=100$ (triangles), $n_s/n_0=1000$ (squares),
dotted line shows
the analytical solution
(Equation \ref{35}) for circles, solid and dashed lines show analytical solutions for
Equation \ref{36} and \ref{37}, respectively, for squares;
(c) $a=10^5 {\rm \, cm}$ (circles), $a=3\times10^5 {\rm \, cm}$ (triangles), $a=10^7 {\rm \, cm}$ (squares),
upper and lower solid lines show analytical solutions for Equation \ref{35} and \ref{37}, respectively;
(d) $n_s/n_0=1000$,
$B_0=1 {\rm \, G}$ (circles), $B_0=10 {\rm \, G}$ (triangles), $B_0=100 {\rm \, G}$ (squares)
dotted line shows analytical solution (Equation \ref{35}) for circles,
solid line shows analytical solution (Equation \ref{36}) for triangles,
dashed line shows analytical solution (Equation \ref{37}) for squares.
}
\label{fig4}
\end{figure}
Equation \ref{32} has two obvious approximations:
${\rm tanh} \, ( k_{y2} a ) \to k_{y2} a$ for small $k_{y2} a$
and ${\rm tanh} \, ( k_{y2} a ) \to 1$ for large $k_{y2} a$.
In what follows, they are called the thin and thick approximations, respectively.
In the first case, the current layer is thin enough, so that the perturbation
arising at one boundary of the layer does not decay
along the way to the other boundary.
On the contrary, the current layer is quite thick
compared to the attenuation length of the perturbation in the second case.
Numerical calculations of Equation \ref{32} with ${\it \Gamma}$ from Equation \ref{33} show that
$k_z^2\ll {\it \Gamma}/\nu_m$ over the entire range of coronal plasma parameters.
Therefore, the dispersion Equation \ref{32} can be simplified.

In the thin approximation,
\begin{equation}
    k_{z{\rm thin}}^2
    \simeq \left[\left( \frac{\tau_\sigma}{\tau_\nu} \,
    \frac{n_0}{n_s}\right)^2 \frac{ 1 }{V_D^2} -
    \frac{ 1 }{ V_S^2 + V_A^2 }\right] \, {\it \Gamma}^2 \, ,
    \label{34}
\end{equation}
where the drift velocity $V_D=\nu_m / a$ is introduced.
This is the velocity at which the plasma drifts into the current layer
(see Section 8.1.1 in \citeauthor{2006ASSL..341.....S}, \citeyear{2006ASSL..341.....S}).
There is no drift in our model, but we will use this notation for convenience.
For sufficiently strong magnetic field (see Equation \ref{20}) and low viscosity,
Equation \ref{34} transforms to
\begin{equation}
    k_{z{\rm thin}}
    \simeq
    \frac{n_0}{n_s} \, \frac{ {\it \Gamma} }{V_D} \, .
    \label{35}
\end{equation}

In the thick approximation,
\begin{equation}
    k_{z{\rm thick}}^2
    \simeq \left( \frac{\tau_\sigma}{\tau_\nu} \,
    \frac{n_0}{n_s}\right)^2 \frac{ {\it \Gamma} }{\nu_m} -
    \frac{ {\it \Gamma}^2 }{ V_S^2 + V_A^2 } \, ,
    \label{36}
\end{equation}
and for strong field and low viscosity,
\begin{equation}
    k_{z{\rm thick}}
    \simeq
    \frac{n_0}{n_s} \, \sqrt{\frac{ {\it \Gamma} }{\nu_m}} \, .
    \label{37}
\end{equation}

Figure \ref{fig4} shows a series of profiles of the dependence
for the spatial period of the instability
calculated as $l=2\pi/k_{z}$ for
$k_z$ from Equation \ref{32} (circles, triangles, and squares)
and $k_{z{\rm thin}}$ and $k_{z{\rm thick}}$ approximations
from Equations \ref{35} and \ref{37}, respectively (thin lines),
on the temperature of the current layer.
The exact calculation of Equations \ref{12} and \ref{32}
is shown by circles, triangles, and squares in the figure.
The approximative Equations \ref{35}\,--\,\ref{37} are shown by solid, dashed, and dotted lines.

The spatial period of the instability
strongly depends on the concentration of the surrounding plasma (Figure \ref{fig4}a).
The graphs are in good agreement with the thin approximation (Equation \ref{35}).
Using Equation \ref{34} instead of Equation \ref{35} does not lead to a visible improvement in the result.
The most remarkable feature of the graphs is a step at $T_s=5\times10^6-10^7 {\rm \, K}$.
The spatial period is constant in a fairly wide temperature range,
and it is this temperature range that seems quite reasonable for a preflare current layer.
It is also reasonable to expect an increase in plasma concentration near the current layer.
With an increase in the strength of the magnetic field
from 1 G (for a quiet corona) to 100 G (for the active region),
the plasma concentration also increases by two orders of magnitude due to magnetic freezing.
Therefore, $n_0=10^{10} {\rm \, cm}^{-3}$ is used for the other graphs in Figure \ref{fig4}.

An increase in the concentration jump does not change the spatial period quantitatively,
but it changes the solution qualitatively (Figure \ref{fig4}b).
Large density jumps at low temperatures $T_s<10^7$ correspond to a thick approximation,
and Equation \ref{36} follows the exact solution much better than Equation \ref{37}.

An increase in the half-thickness of the current layer obviously leads to a thick approximation,
but it also slightly changes the spatial period of instability (Figure \ref{fig4}c).

The influence of the magnetic field is manifested only at high jumps in concentration
when the second term on the right side of the Equations \ref{34} and \ref{36} prevails.
Therefore, Figure \ref{fig4}d is calculated for $n_s/n_0=1000$.
Again, the dependence of the spatial period on the magnitude of the magnetic field is rather weak.
The influence of the magnetic field becomes indistinguishable at lower density contrasts.

As a result, the spatial period of the instability is constant
over a wide range of changes in the parameters
of the coronal plasma at the assumed temperature
of the preflare current layer $T_s=5\times10^6-10^7 {\rm \, K}$
and the concentration of the surrounding plasma $n_0=10^{10} {\rm \, cm}^{-3}$.
Its values belong to in a narrow range from 1 to 10 Mm,
which is in good agreement with the distances between the solar flare loops
observed in the ultraviolet range.

\section{Conclusion}
\label{sec5}

The stability problem of the preflare current layer with respect to small perturbations is addressed.
The problem is solved within the framework of dissipative MHD taking into account
viscosity, electrical and thermal conductivity, and radiative cooling of the plasma.
A piecewise homogeneous current layer model is used.
The simplicity of the model allows one to obtain accurate analytical expressions
for the growth rate (Equation \ref{12}) and the spatial scale (Equation \ref{32}) of instability,
as well as their simple approximations (Equations \ref{33}\,--\,\ref{37}) in the conditions of the solar corona.
The instability has a thermal nature.
It occurs as a result of a drop in thermal conductivity inside the current layer
and increases on the characteristic time scale of radiative plasma cooling.
Due to the structural features of
the radiative loss function
of an optically thin medium,
the spatial instability period is contained in a narrow range of values of about $l=1-10 {\rm \, Mm}$
for a wide range of parameters of the current layer and the surrounding plasma.

\begin{figure}
\begin{center}
\includegraphics*[width=0.5\linewidth]{fig5.eps}
\end{center}
\caption{Location of the perturbed current layer above the arcade of coronal magnetic loops.
The perturbation has a spatial period $l$.
The Roman numbers mark the energy fluxes associated with radiative plasma cooling (I)
and heat conduction (II).}
\label{fig5}
\end{figure}
The instability properties allow us to offer the following qualitative picture of the solar flare triggering.
There is a preflare current layer above the arcade of coronal magnetic loops (Figure \ref{fig5}).
Due to a random perturbation, some of its sections begin to lose more heat by radiation.
High electronic thermal conductivity can redistribute heat between cold and hot areas.
However, if the electronic thermal conductivity is suppressed
by the perturbation of the transverse magnetic field penetrating in the current layer,
then the ionic thermal conductivity does not have time to transfer heat from hot to cold areas.
The temperature difference between the cold and hot sections
of the preflare current layer increases with increment described by Equation \ref{33}.
The alternation of cold and hot sections leads to a wave-like curvature
of the surface of the current layer with a spatial period $l$
due to the total pressure balance.
The curvature has a symmetrical shape in accordance with the solution found.
The current layer begins to disintegrate into individual fibers
located across the direction of the current,
which can lead to its breaking and, as a result, to a solar flare.
The regions of the main energy release will alternate with the same spatial period $l$.
Flows of accelerated charged particles rush into the coronal magnetic loops
located near the regions of energy release, which ultimately leads
to the observed brightening of individual flare loops in the ultraviolet range.

In order to mathematically simplify the model,
many significant physical features of the preflare current layer were neglected.
Magnetic non-neutrality of the current layer leads to
a change in the pressure balance at its boundary, while
the appearance of a component of the magnetic field
normal to the layer changes the type of MHD discontinuity on the boundary
\citep{1985SoPh...95..141S, 1985SoPh..102...79S}.
The finite width of the current layer requires
taking into account the corresponding derivatives with respect to the $ x $ coordinate,
which leads to the appearance of tearing instability
\citep{1988SoPh..117...89S, 1989SoPh..120...93S}.
The observations of flare loops on the Sun indirectly indicate a complex current layer geometry
that is different from a simple planar configuration.
A statistical analysis of the flare loops themselves in the context of the considered model
is a separate complex task.
Attention on these and other issues will be paid in following articles of this series
(Thermal Trigger for Solar Flares).

\begin{acks}
The author thanks Prof. Boris Somov, Vasilisa Nikiforova, and anonymous reviewer for discussing the article.

\noindent{\bf Disclosure of Potential Conflicts of Interest} The author declares that there are no conflicts of interest.
\end{acks}

\bibliographystyle{spr-mp-sola}

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