Sexual dimorphism

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Mandarin ducks, male (left) and female (right), illustrating the dramatic difference in plumage between sexes, a manifestation of sexual dimorphism

Sexual dimorphism is the condition where sexes of the same species exhibit different morphological characteristics, particularly characteristics not directly involved in reproduction. The condition occurs in most animals and some plants. Differences may include secondary sex characteristics, size, weight, color, markings, or behavioral or cognitive traits. Male–male reproductive competition has evolved a diverse array of sexually dimorphic traits. Aggressive utility traits such as "battle" teeth and blunt heads reinforced as battering rams are used as weapons in aggressive interactions between rivals. Passive displays such as ornamental feathering or song-calling have also evolved mainly through sexual selection. These differences may be subtle or exaggerated and may be subjected to sexual selection and natural selection. The opposite of dimorphism is monomorphism, when both biological sexes are phenotypically indistinguishable from each other.

Overview

The peacock, on the right, is courting the peahen, on the left.

Male (bottom) and female mallards. The male mallard has an unmistakable bottle green head when his breeding plumage is present.

Ornamentation and coloration

Orgyia antiqua male (left) and female (right).

Common and easily identified types of dimorphism consist of ornamentation and coloration, though not always apparent. A difference in the coloration of sexes within a given species is called sexual dichromatism, commonly seen in many species of birds and reptiles. Sexual selection leads to the exaggerated dimorphic traits that are used predominantly in competition over mates. The increased fitness resulting from ornamentation offsets its cost to produce or maintain suggesting complex evolutionary implications, but the costs and evolutionary implications vary from species to species. The costs and implications differ depending on the nature of the ornamentation (such as the color mechanism involved).

The peafowl constitute conspicuous illustrations of the principle. The ornate plumage of peacocks, as used in the courting display, attracts peahens. At first sight, one might mistake peacocks and peahens for completely different species because of the vibrant colours and the sheer size of the male's plumage; the peahen is of a subdued brown coloration. The plumage of the peacock increases its vulnerability to predators because it is a hindrance in flight, and it renders the bird conspicuous in general. Similar examples are manifold, such as in birds of paradise and argus pheasants.

Another example of sexual dichromatism is that of the nestling blue tits. Males are chromatically more yellow than females. It is believed that this is obtained by the ingestion of green Lepidopteran larvae, which contain large amounts of the carotenoids lutein and zeaxanthin. This diet also affects the sexually dimorphic colours in the human-invisible ultraviolet spectrum. Hence, the male birds, although appearing yellow to humans actually have a violet-tinted plumage that is seen by females. This plumage is thought to be an indicator of male parental abilities. Perhaps this is a good indicator for females because it shows that they are good at obtaining a food supply from which the carotenoid is obtained. There is a positive correlation between the chromas of the tail and breast feathers and body condition. Carotenoids play an important role in immune function for many animals, so carotenoid dependent signals might indicate health.

Frogs constitute another conspicuous illustration of the principle. There are two types of dichromatism for frog species: ontogenetic and dynamic. Ontogenetic frogs are more common and have permanent color changes in males or females. Ranoidea lesueuri is an example of a dynamic frog with temporary color changes in males during the breeding season. Hyperolius ocellatus is an ontogenetic frog with dramatic differences in both color and pattern between the sexes. At sexual maturity, the males display a bright green with white dorsolateral lines. In contrast, the females are rusty red to silver with small spots. The bright coloration in the male population attracts females and is an aposematic sign to potential predators.

Females often show a preference for exaggerated male secondary sexual characteristics in mate selection. The sexy son hypothesis explains that females prefer more elaborate males and select against males that are dull in color, independent of the species' vision.

Similar sexual dimorphism and mating choice are also observed in many fish species. For example, male guppies have colorful spots and ornamentations, while females are generally grey. Female guppies prefer brightly colored males to duller males.

In redlip blennies, only the male fish develops an organ at the anal-urogenital region that produces antimicrobial substances. During parental care, males rub their anal-urogenital regions over their nests' internal surfaces, thereby protecting their eggs from microbial infections, one of the most common causes for mortality in young fish.

Plants

Most flowering plants are hermaphroditic but approximately 6% of species have separate males and females (dioecy). Sexual dimorphism is common in dioecious plants and dioicous species.

Males and females in insect-pollinated species generally look similar to one another because plants provide rewards (e.g. nectar) that encourage pollinators to visit another similar flower, completing pollination. Catasetum orchids are one interesting exception to this rule. Male Catasetum orchids violently attach pollinia to euglossine bee pollinators. The bees will then avoid other male flowers but may visit the female, which looks different from the males.

Various other dioecious exceptions, such as Loxostylis alata have visibly different sexes, with the effect of eliciting the most efficient behavior from pollinators, who then use the most efficient strategy in visiting each gender of flower instead of searching, say, for pollen in a nectar-bearing female flower.

Some plants, such as some species of Geranium have what amounts to serial sexual dimorphism. The flowers of such species might, for example, present their anthers on opening, then shed the exhausted anthers after a day or two and perhaps change their colours as well while the pistil matures; specialist pollinators are very much inclined to concentrate on the exact appearance of the flowers they serve, which saves their time and effort and serves the interests of the plant accordingly. Some such plants go even further and change their appearance once fertilized, thereby discouraging further visits from pollinators. This is advantageous to both parties because it avoids damaging the developing fruit and wasting the pollinator's effort on unrewarding visits. In effect, the strategy ensures that pollinators can expect a reward every time they visit an appropriately advertising flower.

Females of the aquatic plant Vallisneria americana have floating flowers attached by a long flower stalk that are fertilized if they contact one of the thousands of free-floating flowers released by a male. Sexual dimorphism is most often associated with wind-pollination in plants due to selection for efficient pollen dispersal in males vs pollen capture in females, e.g. Leucadendron rubrum.

Sexual dimorphism in plants can also be dependent on reproductive development. This can be seen in Cannabis sativa, a type of hemp, which have higher photosynthesis rates in males while growing but higher rates in females once the plants become sexually mature.

Every sexually reproducing extant species of the vascular plant has an alternation of generations; the plants we see about us generally are diploid sporophytes, but their offspring are not the seeds that people commonly recognize as the new generation. The seed actually is the offspring of the haploid generation of microgametophytes (pollen) and megagametophytes (the embryo sacs in the ovules). Each pollen grain accordingly may be seen as a male plant in its own right; it produces a sperm cell and is dramatically different from the female plant, the megagametophyte that produces the female gamete.

Insects

Colias dimera mating. The male is a brighter yellow than the female.

Insects display a wide variety of sexual dimorphism between taxa including size, ornamentation and coloration. The female-biased sexual size dimorphism observed in many taxa evolved despite intense male–male competition for mates. In Osmia rufa, for example, the female is larger/broader than males, with males being 8–10 mm in size and females being 10–12 mm in size. In the hackberry emperor females are similarly larger than males. The reason for the sexual dimorphism is due to provision size mass, in which females consume more pollen than males.

In some species, there is evidence of male dimorphism, but it appears to be for distinctions of roles. This is seen in the bee species Macrotera portalis in which there is a small-headed morph, capable of flight, and large-headed morph, incapable of flight, for males. Anthidium manicatum also displays male-biased sexual dimorphism. The selection for larger size in males rather than females in this species may have resulted due to their aggressive territorial behavior and subsequent differential mating success. Another example is Lasioglossum hemichalceum, which is a species of sweat bee that shows drastic physical dimorphisms between male offspring. Not all dimorphism has to have a drastic difference between the sexes. Andrena agilissima is a mining bee where the females only have a slightly larger head than the males.

Weaponry leads to increased fitness by increasing success in male–male competition in many insect species. The beetle horns in Onthophagus taurus are enlarged growths of the head or thorax expressed only in the males. Copris ochus also has distinct sexual and male dimorphism in head horns. These structures are impressive because of the exaggerated sizes. There is a direct correlation between male horn lengths and body size and higher access to mates and fitness. In other beetle species, both males and females may have ornamentation such as horns. Generally, insect sexual size dimorphism (SSD) within species increases with body size.

Sexual dimorphism within insects is also displayed by dichromatism. In butterfly genera Bicyclus and Junonia, dimorphic wing patterns evolved due to sex-limited expression, which mediates the intralocus sexual conflict and leads to increased fitness in males. The sexual dichromatic nature of Bicyclus anynana is reflected by female selection on the basis of dorsal UV-reflective eyespot pupils. The common brimstone also displays sexual dichromatism; males have yellow and iridescent wings, while female wings are white and non-iridescent. Naturally selected deviation in protective female coloration is displayed in mimetic butterflies.

Spiders and sexual cannibalism

Female (left) and male (right) Argiope appensa, displaying typical sexual differences in spiders, with dramatically smaller males

Hammock Spiders (Pityohyphantes sp.) courting. Female left and male right.

Many arachnid groups exhibit sexual dimorphism, but it is most widely studied in the spiders. In the orb-weaving spider Zygiella x-notata, for example, adult females have a larger body size than adult males. Size dimorphism shows a correlation with sexual cannibalism, which is prominent in spiders (it is also found in insects such as praying mantises). In the size dimorphic wolf spider Tigrosa helluo, food-limited females cannibalize more frequently. Therefore, there is a high risk of low fitness for males due to pre-copulatory cannibalism, which led to male selection of larger females for two reasons: higher fecundity and lower rates of cannibalism. In addition, female fecundity is positively correlated with female body size and large female body size is selected for, which is seen in the family Araneidae. All Argiope species, including Argiope bruennichi, use this method. Some males evolved ornamentation including binding the female with silk, having proportionally longer legs, modifying the female's web, mating while the female is feeding, or providing a nuptial gift in response to sexual cannibalism. Male body size is not under selection due to cannibalism in all spider species such as Nephila pilipes, but is more prominently selected for in less dimorphic species of spiders, which often selects for larger male size. In the species Maratus volans, the males are known for their characteristic colorful fan which attracts the females during mating.

Fish

Ray finned fish are an ancient and diverse class, with the widest degree of sexual dimorphism of any animal class. Fairbairn notes that "females are generally larger than males but males are often larger in species with male–male combat or male paternal care ... [sizes range] from dwarf males to males more than 12 times heavier than females."

There are cases where males are substantially larger than females. An example is Lamprologus callipterus, a type of cichlid fish. In this fish, the males are characterized as being up to 60 times larger than the females. The male's increased size is believed to be advantageous because males collect and defend empty snail shells in each of which a female breeds. Males must be larger and more powerful in order to collect the largest shells. The female's body size must remain small because in order for her to breed, she must lay her eggs inside the empty shells. If she grows too large, she will not fit in the shells and will be unable to breed. The female's small body size is also likely beneficial to her chances of finding an unoccupied shell. Larger shells, although preferred by females, are often limited in availability. Hence, the female is limited to the growth of the size of the shell and may actually change her growth rate according to shell size availability. In other words, the male's ability to collect large shells depends on his size. The larger the male, the larger the shells he is able to collect. This then allows for females to be larger in his brooding nest which makes the difference between the sizes of the sexes less substantial. Male–male competition in this fish species also selects for large size in males. There is aggressive competition by males over territory and access to larger shells. Large males win fights and steal shells from competitors. Another example is the dragonet, in which males are considerably larger than females and possess longer fins.

Sexual dimorphism also occurs in hermaphroditic fish. These species are known as sequential hermaphrodites. In fish, reproductive histories often include the sex-change from female to male where there is a strong connection between growth, the sex of an individual, and the mating system it operates within. In protogynous mating systems where males dominate mating with many females, size plays a significant role in male reproductive success. Males have a propensity to be larger than females of a comparable age but it is unclear whether the size increase is due to a growth spurt at the time of the sexual transition or due to the history of faster growth in sex changing individuals. Larger males are able to stifle the growth of females and control environmental resources.

Social organization plays a large role in the changing of sex by the fish. It is often seen that a fish will change its sex when there is a lack of dominant male within the social hierarchy. The females that change sex are often those who attain and preserve an initial size advantage early in life. In either case, females which change sex to males are larger and often prove to be a good example of dimorphism.

In other cases with fish, males will go through noticeable changes in body size, and females will go through morphological changes that can only be seen inside of the body. For example, in sockeye salmon, males develop larger body size at maturity, including an increase in body depth, hump height, and snout length. Females experience minor changes in snout length, but the most noticeable difference is the huge increase in gonad size, which accounts for about 25% of body mass.

Sexual selection was observed for female ornamentation in Gobiusculus flavescens, known as two-spotted gobies. Traditional hypotheses suggest that male–male competition drives selection. However, selection for ornamentation within this species suggests that showy female traits can be selected through either female–female competition or male mate choice. Since carotenoid-based ornamentation suggests mate quality, female two-spotted guppies that develop colorful orange bellies during the breeding season are considered favorable to males. The males invest heavily in offspring during the incubation, which leads to the sexual preference in colorful females due to higher egg quality.

Amphibians and non-avian reptiles

Mississippi map turtles (Graptemys pseudogeographica kohni) adult female (left) and adult male (right)

In amphibians and reptiles, the degree of sexual dimorphism varies widely among taxonomic groups. The sexual dimorphism in amphibians and reptiles may be reflected in any of the following: anatomy; relative length of tail; relative size of head; overall size as in many species of vipers and lizards; coloration as in many amphibians, snakes, and lizards, as well as in some turtles; an ornament as in many newts and lizards; the presence of specific sex-related behaviour is common to many lizards; and vocal qualities which are frequently observed in frogs.

Anole lizards show prominent size dimorphism with males typically being significantly larger than females. For instance, the average male Anolis sagrei was 53.4 mm vs. 40 mm in females. Different sizes of the heads in anoles have been explained by differences in the estrogen pathway. The sexual dimorphism in lizards is generally attributed to the effects of sexual selection, but other mechanisms including ecological divergence and fecundity selection provide alternative explanations. The development of color dimorphism in lizards is induced by hormonal changes at the onset of sexual maturity, as seen in Psamodromus algirus, Sceloporus gadoviae, and S. undulates erythrocheilus. Sexual dimorphism in size is also seen in frog species like P. bibronii.

Male painted dragon lizards, Ctenophorus pictus. are brightly conspicuous in their breeding coloration, but male colour declines with aging. Male coloration appears to reflect innate anti-oxidation capacity that protects against oxidative DNA damage. Male breeding coloration is likely an indicator to females of the underlying level of oxidative DNA damage (a significant component of aging) in potential mates.

Birds

Female (left) and male (right) common pheasant, showing that the male is much larger and more colorful than the female

Some bird species, such as this mute swan, do not display sexual dimorphism through their plumage, and instead can be distinguished by other physiological or behavioural characteristics. Generally, male Mute swans, or cobs, are taller and larger than females, or pens, and have thicker necks and a more pronounced 'knob' above their bill.

Skeletons of female (left) and Male (right) black-casqued hornbills (Ceratogymna atrata). The difference between the sexes is apparent in the casque on the top of their bill. This pair is on display at the Museum of Osteology.

The eclectus parrot is a rare example of a bird where the female (right) is more colorful than the male (left)

Sexual dimorphism in birds can be manifested in size or plumage differences between the sexes. Sexual size dimorphism varies among taxa with males typically being larger, though this is not always the case, e.g. birds of prey, hummingbirds, and some species of flightless birds. Plumage dimorphism, in the form of ornamentation or coloration, also varies, though males are typically the more ornamented or brightly colored sex. Such differences have been attributed to the unequal reproductive contributions of the sexes. This difference produces a stronger female choice since they have more risk in producing offspring. In some species, the male's contribution to reproduction ends at copulation, while in other species the male becomes the main (or only) caregiver. Plumage polymorphisms have evolved to reflect these differences and other measures of reproductive fitness, such as body condition or survival. The male phenotype sends signals to females who then choose the 'fittest' available male.

Sexual dimorphism is a product of both genetics and environmental factors. An example of sexual polymorphism determined by environmental conditions exists in the red-backed fairywren. Red-backed fairywren males can be classified into three categories during breeding season: black breeders, brown breeders, and brown auxiliaries. These differences arise in response to the bird's body condition: if they are healthy they will produce more androgens thus becoming black breeders, while less healthy birds produce less androgens and become brown auxiliaries. The reproductive success of the male is thus determined by his success during each year's non-breeding season, causing reproductive success to vary with each year's environmental conditions.

Migratory patterns and behaviors also influence sexual dimorphisms. This aspect also stems back to the size dimorphism in species. It has been shown that the larger males are better at coping with the difficulties of migration and thus are more successful in reproducing when reaching the breeding destination. When viewing this in an evolutionary standpoint many theories and explanations come to consideration. If these are the result for every migration and breeding season the expected results should be a shift towards a larger male population through sexual selection. Sexual selection is strong when the factor of environmental selection is also introduced. The environmental selection may support a smaller chick size if those chicks were born in an area that allowed them to grow to a larger size, even though under normal conditions they would not be able to reach this optimal size for migration. When the environment gives advantages and disadvantages of this sort, the strength of selection is weakened and the environmental forces are given greater morphological weight. The sexual dimorphism could also produce a change in timing of migration leading to differences in mating success within the bird population. When the dimorphism produces that large of a variation between the sexes and between the members of the sexes multiple evolutionary effects can take place. This timing could even lead to a speciation phenomenon if the variation becomes strongly drastic and favorable towards two different outcomes. Sexual dimorphism is maintained by the counteracting pressures of natural selection and sexual selection. For example, sexual dimorphism in coloration increases the vulnerability of bird species to predation by European sparrowhawks in Denmark. Presumably, increased sexual dimorphism means males are brighter and more conspicuous, leading to increased predation. Moreover, the production of more exaggerated ornaments in males may come at the cost of suppressed immune function. So long as the reproductive benefits of the trait due to sexual selection are greater than the costs imposed by natural selection, then the trait will propagate throughout the population. Reproductive benefits arise in the form of a larger number of offspring, while natural selection imposes costs in the form of reduced survival. This means that even if the trait causes males to die earlier, the trait is still beneficial so long as males with the trait produce more offspring than males lacking the trait. This balance keeps the dimorphism alive in these species and ensures that the next generation of successful males will also display these traits that are attractive to the females.

Such differences in form and reproductive roles often cause differences in behavior. As previously stated, males and females often have different roles in reproduction. The courtship and mating behavior of males and females are regulated largely by hormones throughout a bird's lifetime. Activational hormones occur during puberty and adulthood and serve to 'activate' certain behaviors when appropriate, such as territoriality during breeding season. Organizational hormones occur only during a critical period early in development, either just before or just after hatching in most birds, and determine patterns of behavior for the rest of the bird's life. Such behavioral differences can cause disproportionate sensitivities to anthropogenic pressures. Females of the whinchat in Switzerland breed in intensely managed grasslands. Earlier harvesting of the grasses during the breeding season lead to more female deaths. Populations of many birds are often male-skewed and when sexual differences in behavior increase this ratio, populations decline at a more rapid rate. Also not all male dimorphic traits are due to hormones like testosterone, instead they are a naturally occurring part of development, for example plumage. In addition, the strong hormonal influence on phenotypic differences suggest that the genetic mechanism and genetic basis of these sexually dimorphic traits may involve transcription factors or cofactors rather than regulatory sequences.

Sexual dimorphism may also influence differences in parental investment during times of food scarcity. For example, in the blue-footed booby, the female chicks grow faster than the males, resulting in booby parents producing the smaller sex, the males, during times of food shortage. This then results in the maximization of parental lifetime reproductive success. In Black-tailed Godwits Limosa limosa limosa females are also the larger sex, and the growth rates of female chicks are more susceptible to limited environmental conditions.

Sexual dimorphism may also only appear during mating season, some species of birds only show dimorphic traits in seasonal variation. The males of these species will molt into a less bright or less exaggerated color during the off breeding season. This occurs because the species is more focused on survival than reproduction, causing a shift into a less ornate state.

Consequently, sexual dimorphism has important ramifications for conservation. However, sexual dimorphism is not only found in birds and is thus important to the conservation of many animals. Such differences in form and behavior can lead to sexual segregation, defined as sex differences in space and resource use. Most sexual segregation research has been done on ungulates, but such research extends to bats, kangaroos, and birds. Sex-specific conservation plans have even been suggested for species with pronounced sexual segregation.

The term sesquimorphism (the Latin numeral prefix sesqui- means one-and-one-half, so halfway between mono- (one) and di- (two)) has been proposed for bird species in which "both sexes have basically the same plumage pattern, though the female is clearly distinguishable by reason of her paler or washed-out colour". Examples include Cape sparrow (Passer melanurus), rufous sparrow (subspecies P. motinensis motinensis), and saxaul sparrow (P. ammodendri).

Mammals

In a large proportion of mammal species, males are larger than females. Both genes and hormones affect the formation of many animal brains before "birth" (or hatching), and also behaviour of adult individuals. Hormones significantly affect human brain formation, and also brain development at puberty. A 2004 review in Nature Reviews Neuroscience observed that "because it is easier to manipulate hormone levels than the expression of sex chromosome genes, the effects of hormones have been studied much more extensively, and are much better understood, than the direct actions in the brain of sex chromosome genes." It concluded that while "the differentiating effects of gonadal secretions seem to be dominant," the existing body of research "support the idea that sex differences in neural expression of X and Y genes significantly contribute to sex differences in brain functions and disease."

Pinnipeds

Male and female northern elephant seal, the male being larger with a big proboscis

Marine mammals show some of the greatest sexual size differences of mammals, because of sexual selection and environmental factors like breeding location. The mating system of pinnipeds varies from polygamy to serial monogamy. Pinnipeds are known for early differential growth and maternal investment since the only nutrients for newborn pups is the milk provided by the mother. For example, the males are significantly larger (about 10% heavier and 2% longer) than the females at birth in sea lion pups. The pattern of differential investment can be varied principally prenatally and post-natally. Mirounga leonina, the southern elephant seal, is one of the most dimorphic mammals.

Primates

Main article: Sexual dimorphism in non-human primates

Humans

Main articles: Sex differences in humans and Sex differences in human psychology

Top: Stylised illustration of humans on the Pioneer plaque, showing both male (left) and female (right). Bottom: Comparison between male (left) and female (right) pelvises.

According to Clark Spencer Larsen, modern day Homo sapiens show a range of sexual dimorphism, with average body mass between the sexes differing by roughly 15%. Substantial discussion in academic literature considers potential evolutionary advantages associated with sexual competition (both intrasexual and intersexual), as well as short- and long-term sexual strategies. According to Daly and Wilson, "The sexes differ more in human beings than in monogamous mammals, but much less than in extremely polygamous mammals."

The average basal metabolic rate is about 6 percent higher in adolescent males than females and increases to about 10 percent higher after puberty. Females tend to convert more food into fat, while males convert more into muscle and expendable circulating energy reserves. Aggregated data of absolute strength indicates that females have, on average, 40–60% the upper body strength of males, and 70–75% the lower body strength. The difference in strength relative to body mass is less pronounced in trained individuals. In Olympic weightlifting, male records vary from 5.5× body mass in the lowest weight category to 4.2× in the highest weight category, while female records vary from 4.4× to 3.8×, a weight adjusted difference of only 10–20%, and an absolute difference of about 40% (i.e. 472 kg vs 333 kg for unlimited weight classes; see Olympic weightlifting records). A study, carried about by analyzing annual world rankings from 1980 to 1996, found that males' running times were, on average, 11% faster than females'.

In early adolescence, females are on average taller than males (as females tend to go through puberty earlier), but males, on average, surpass them in height in later adolescence and adulthood. In the United States, adult males are on average 9% taller and 16.5% heavier than adult females.

Males typically have larger tracheae and branching bronchi, with about 30 percent greater lung volume per body mass. On average, males have larger hearts, 10 percent higher red blood cell count, higher hemoglobin, hence greater oxygen-carrying capacity. They also have higher circulating clotting factors (vitamin K, prothrombin and platelets). These differences lead to faster healing of wounds and lower sensitivity to nerve pain after injury. In males, pain-causing injury to the peripheral nerve occurs through the microglia, while in females it occurs through the T cells (except in pregnant women, who follow a male pattern).

Females typically have more white blood cells (stored and circulating), as well as more granulocytes and B and T lymphocytes. Additionally, they produce more antibodies at a faster rate than males, hence they develop fewer infectious diseases and succumb for shorter periods. Ethologists argue that females, interacting with other females and multiple offspring in social groups, have experienced such traits as a selective advantage. Females have a higher sensitivity to pain due to aforementioned nerve differences that increase the sensation, and females thus require higher levels of pain medication after injury. Hormonal changes in females affect pain sensitivity, and pregnant women have the same sensitivity as males. Acute pain tolerance is also more consistent over a lifetime in females than males, despite these hormonal changes. Despite differences in the physical feeling, both sexes have similar psychological tolerance to (or ability to cope with and ignore) pain.

In the human brain, a difference between sexes was observed in the transcription of the PCDH11X/Y gene pair unique to Homo sapiens. Sexual differentiation in the human brain from the undifferentiated state is triggered by testosterone from the fetal testis. Testosterone is converted to estrogen in the brain through the action of the enzyme aromatase. Testosterone acts on many brain areas, including the SDN-POA, to create the masculinized brain pattern. Brains of pregnant females carrying male fetuses may be shielded from the masculinizing effects of androgen through the action of sex hormone-binding globulin.

The relationship between sex differences in the brain and human behavior is a subject of controversy in psychology and society at large. Many females tend to have a higher ratio of gray matter in the left hemisphere of the brain in comparison to males. Males on average have larger brains than females; however, when adjusted for total brain volume the gray matter differences between sexes is almost nonexistent. Thus, the percentage of gray matter appears to be more related to brain size than it is to sex. Differences in brain physiology between sexes do not necessarily relate to differences in intellect. Haier et al. found in a 2004 study that "men and women apparently achieve similar IQ results with different brain regions, suggesting that there is no singular underlying neuroanatomical structure to general intelligence and that different types of brain designs may manifest equivalent intellectual performance". (See the sex and intelligence article for more on this subject.) Strict graph-theoretical analysis of the human brain connections revealed that in numerous graph-theoretical parameters (e.g., minimum bipartition width, edge number, the expander graph property, minimum vertex cover), the structural connectome of women are significantly "better" connected than the connectome of men. It was shown that the graph-theoretical differences are due to the sex and not to the differences in the cerebral volume, by analyzing the data of 36 females and 36 males, where the brain volume of each man in the group was smaller than the brain volume of each woman in the group.

Sexual dimorphism was also described in the gene level and shown to extend from the sex chromosomes. Overall, about 6500 genes have been found to have sex-differential expression in at least one tissue. Many of these genes are not directly associated with reproduction, but rather linked to more general biological features. In addition, it has been shown that genes with sex-specific expression undergo reduced selection efficiency, which lead to higher population frequencies of deleterious mutations and contributing to the prevalence of several human diseases.

Immune function

Sexual dimorphism in immune function is a common pattern in vertebrates and also in a number of invertebrates. Most often, females are more 'immunocompetent' than males. This trait is not consistent among all animals, but differs depending on taxonomy, with the most female-biased immune systems being found in insects. In mammals this results in more frequent and severe infections in males and higher rates of autoimmune disorders in females. One potential cause may be differences in gene expression of immune cells between the sexes. Another explanation is that endocrinological differences between the sexes impact the immune system – for example, testosterone acts as an immunosuppressive agent.

Cells

Phenotypic differences between sexes are evident even in cultured cells from tissues. For example, female muscle-derived stem cells have a better muscle regeneration efficiency than male ones. There are reports of several metabolic differences between male and female cells and they also respond to stress differently.

Reproductively advantageous

In theory, larger females are favored by competition for mates, especially in polygamous species. Larger females offer an advantage in fertility, since the physiological demands of reproduction are limiting in females. Hence there is a theoretical expectation that females tend to be larger in species that are monogamous. Females are larger in many species of insects, many spiders, many fish, many reptiles, owls, birds of prey and certain mammals such as the spotted hyena, and baleen whales such as blue whale. As an example, in some species, females are sedentary, and so males must search for them. Fritz Vollrath and Geoff Parker argue that this difference in behaviour leads to radically different selection pressures on the two sexes, evidently favouring smaller males. Cases where the male is larger than the female have been studied as well, and require alternative explanations.

One example of this type of sexual size dimorphism is the bat Myotis nigricans, (black myotis bat) where females are substantially larger than males in terms of body weight, skull measurement, and forearm length. The interaction between the sexes and the energy needed to produce viable offspring make it favorable for females to be larger in this species. Females bear the energetic cost of producing eggs, which is much greater than the cost of making sperm by the males. The fecundity advantage hypothesis states that a larger female is able to produce more offspring and give them more favorable conditions to ensure their survival; this is true for most ectotherms. A larger female can provide parental care for a longer time while the offspring matures. The gestation and lactation periods are fairly long in M. nigricans, the females suckling their offspring until they reach nearly adult size. They would not be able to fly and catch prey if they did not compensate for the additional mass of the offspring during this time. Smaller male size may be an adaptation to increase maneuverability and agility, allowing males to compete better with females for food and other resources.

Female triplewart seadevil, an anglerfish, with male attached near vent (arrow)

Some species of anglerfish also display extreme sexual dimorphism. Females are more typical in appearance to other fish, whereas the males are tiny rudimentary creatures with stunted digestive systems. A male must find a female and fuse with her: he then lives parasitically, becoming little more than a sperm-producing body in what amounts to an effectively hermaphrodite composite organism. A similar situation is found in the Zeus water bug Phoreticovelia disparata where the female has a glandular area on her back that can serve to feed a male, which clings to her (although males can survive away from females, they generally are not free-living). This is taken to the logical extreme in the Rhizocephala crustaceans, like the Sacculina, where the male injects itself into the female's body and becomes nothing more than sperm producing cells, to the point that the superorder used to be mistaken for hermaphroditic.

Some plant species also exhibit dimorphism in which the females are significantly larger than the males, such as in the moss Dicranum and the liverwort Sphaerocarpos. There is some evidence that, in these genera, the dimorphism may be tied to a sex chromosome, or to chemical signalling from females.

Another complicated example of sexual dimorphism is in Vespula squamosa, the southern yellowjacket. In this wasp species, the female workers are the smallest, the male workers are slightly larger, and the female queens are significantly larger than her female worker and male counterparts.

Evolution

See also: Sexual selection and Mate choice

Sexual dimorphism in Cambrian trilobites.

In 1871, Charles Darwin advanced the theory of sexual selection, which related sexual dimorphism with sexual selection.

The first step towards sexual dimorphism is the size differentiation of sperm and eggs (anisogamy). Anisogamy and the usually large number of small male gametes relative to the larger female gametes usually lies in the development of strong sperm competition, because small sperm enable organisms to produce a large number of sperm, and make males (or male function of hermaphrodites) more redundant.

This intensifies male competition for mates and promotes the evolution of other sexual dimorphism in many species, especially in vertebrates including mammals. However, in some species females compete for mates in ways more usually associated with males (usually species in which males invest a lot in rearing offspring and thus are no longer considered as so redundant).

Sexual dimorphism by size is evident in some extinct species such as the velociraptor. In the case of velociraptors the sexual size dimorphism may have been caused by two factors: male competition for hunting ground to attract mates, and/or female competition for nesting locations and mates, males being a scarce breeding resource.

Volvocine algae have been useful in understanding the evolution of sexual dimorphism and species like the beetle C. maculatus, where the females are larger than the males, are used to study its underlying genetic mechanisms. 

In many non-monogamous species, the benefit to a male's reproductive fitness of mating with multiple females is large, whereas the benefit to a female's reproductive fitness of mating with multiple males is small or nonexistent. In these species, there is a selection pressure for whatever traits enable a male to have more matings. The male may therefore come to have different traits from the female.

Male (left), offspring, and female (right) Sumatran orangutans.

These traits could be ones that allow him to fight off other males for control of territory or a harem, such as large size or weapons; or they could be traits that females, for whatever reason, prefer in mates. Male–male competition poses no deep theoretical questions but mate choice does.

Females may choose males that appear strong and healthy, thus likely to possess "good alleles" and give rise to healthy offspring. In some species, however, females seem to choose males with traits that do not improve offspring survival rates, and even traits that reduce it (potentially leading to traits like the peacock's tail). Two hypotheses for explaining this fact are the sexy son hypothesis and the handicap principle.

The sexy son hypothesis states that females may initially choose a trait because it improves the survival of their young, but once this preference has become widespread, females must continue to choose the trait, even if it becomes harmful. Those that do not will have sons that are unattractive to most females (since the preference is widespread) and so receive few matings.

The handicap principle states that a male who survives despite possessing some sort of handicap thus proves that the rest of his genes are "good alleles". If males with "bad alleles" could not survive the handicap, females may evolve to choose males with this sort of handicap; the trait is acting as a hard-to-fake signal of fitness.

Turbulence

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Turbulence

For the turbulence felt on an airplane, see Clear-air turbulence. For other uses, see Turbulence (disambiguation).

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe.

The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Richard Feynman described turbulence as the most important unsolved problem in classical physics.

The turbulence intensity affects many fields, for examples fish ecology, air pollution, precipitation, and climate change.

Examples of turbulence

Laminar and turbulent water flow over the hull of a submarine. As the relative velocity of the water increases turbulence occurs.

Turbulence in the tip vortex from an airplane wing passing through coloured smoke

• Smoke rising from a cigarette. For the first few centimeters, the smoke is laminar. The smoke plume becomes turbulent as its Reynolds number increases with increases in flow velocity and characteristic length scale.
• Flow over a golf ball. (This can be best understood by considering the golf ball to be stationary, with air flowing over it.) If the golf ball were smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the boundary layer would separate early, as the pressure gradient switched from favorable (pressure decreasing in the flow direction) to unfavorable (pressure increasing in the flow direction), creating a large region of low pressure behind the ball that creates high form drag. To prevent this, the surface is dimpled to perturb the boundary layer and promote turbulence. This results in higher skin friction, but it moves the point of boundary layer separation further along, resulting in lower drag.
• Clear-air turbulence experienced during airplane flight, as well as poor astronomical seeing (the blurring of images seen through the atmosphere).
• Most of the terrestrial atmospheric circulation.
• The oceanic and atmospheric mixed layers and intense oceanic currents.
• The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas scrubbers, dynamic scraped surface heat exchangers, etc.) and machines (for instance, internal combustion engines and gas turbines).
• The external flow over all kinds of vehicles such as cars, airplanes, ships, and submarines.
• The motions of matter in stellar atmospheres.
• A jet exhausting from a nozzle into a quiescent fluid. As the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created. These layers separate the fast moving jet from the external fluid, and at a certain critical Reynolds number they become unstable and break down to turbulence.
• Biologically generated turbulence resulting from swimming animals affects ocean mixing.
• Snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence.
• Bridge supports (piers) in water. When river flow is slow, water flows smoothly around the support legs. When the flow is faster, a higher Reynolds number is associated with the flow. The flow may start off laminar but is quickly separated from the leg and becomes turbulent.
• In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structures and turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence. The turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, and they play a critical role in terms of sediment scour, accretion and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, and in the atmosphere.

Unsolved problem in physics:

Is it possible to make a theoretical model to describe the behavior of a turbulent flow—in particular, its internal structures?

• In the medical field of cardiology, a stethoscope is used to detect heart sounds and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.
• Recently, turbulence in porous media became a highly debated subject.
• Strategies used by animals for olfactory navigation, and their success, are heavily influenced by turbulence affecting the odor plume.

Features

Flow visualization of a turbulent jet, made by laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.

Turbulence is characterized by the following features:

Irregularity

Turbulent flows are always highly irregular. For this reason, turbulence problems are normally treated statistically rather than deterministically. Turbulent flow is chaotic. However, not all chaotic flows are turbulent.

Diffusivity

The readily available supply of energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is called "diffusivity".

Turbulent diffusion is usually described by a turbulent diffusion coefficient. This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, and not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes a constitutive relation between a turbulent flux and the gradient of a mean variable similar to the relation between flux and gradient that exists for molecular transport. In the best case, this assumption is only an approximation. Nevertheless, the turbulent diffusivity is the simplest approach for quantitative analysis of turbulent flows, and many models have been postulated to calculate it. For instance, in large bodies of water like oceans this coefficient can be found using Richardson's four-third power law and is governed by the random walk principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula.

Rotationality

Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching. In fluid dynamics, they are essentially vortices subjected to stretching associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. On the other hand, vortex stretching is the core mechanism on which the turbulence energy cascade relies to establish and maintain identifiable structure function. In general, the stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements. As a result, the radial length scale of the vortices decreases and the larger flow structures break down into smaller structures. The process continues until the small scale structures are small enough that their kinetic energy can be transformed by the fluid's molecular viscosity into heat. Turbulent flow is always rotational and three dimensional. For example, atmospheric cyclones are rotational but their substantially two-dimensional shapes do not allow vortex generation and so are not turbulent. On the other hand, oceanic flows are dispersive but essentially non rotational and therefore are not turbulent.

Dissipation

To sustain turbulent flow, a persistent source of energy supply is required because turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies of many different length scales. Most of the kinetic energy of the turbulent motion is contained in the large-scale structures. The energy "cascades" from these large-scale structures to smaller scale structures by an inertial and essentially inviscid mechanism. This process continues, creating smaller and smaller structures which produces a hierarchy of eddies. Eventually this process creates structures that are small enough that molecular diffusion becomes important and viscous dissipation of energy finally takes place. The scale at which this happens is the Kolmogorov length scale.

Via this energy cascade, turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a mean flow. The eddies are loosely defined as coherent patterns of flow velocity, vorticity and pressure. Turbulent flows may be viewed as made of an entire hierarchy of eddies over a wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in flow velocity fluctuations for each length scale (wavenumber). The scales in the energy cascade are generally uncontrollable and highly non-symmetric. Nevertheless, based on these length scales these eddies can be divided into three categories.

Integral time scale

The integral time scale for a Lagrangian flow can be defined as:

${\displaystyle T=\left(\frac{1}{⟨u^{\prime }{u}^{\prime }⟩}\right){\int }_0^{\mathrm{\infty }}⟨u^{\prime }{u}^{\prime }(\tau )⟩d\tau }$

where u′ is the velocity fluctuation, and ${\displaystyle \tau }$ is the time lag between measurements.

Integral length scales

Large eddies obtain energy from the mean flow and also from each other. Thus, these are the energy production eddies which contain most of the energy. They have the large flow velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point flow velocity correlations. The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundreds kilometers.: The integral length scale can be defined as

${\displaystyle L=\left(\frac{1}{⟨u^{\prime }{u}^{\prime }⟩}\right){\int }_0^{\mathrm{\infty }}⟨u^{\prime }{u}^{\prime }(r)⟩dr}$

where r is the distance between two measurement locations, and u′ is the velocity fluctuation in that same direction.

Kolmogorov length scales

Smallest scales in the spectrum that form the viscous sub-layer range. In this range, the energy input from nonlinear interactions and the energy drain from viscous dissipation are in exact balance. The small scales have high frequency, causing turbulence to be locally isotropic and homogeneous.

Taylor microscales

The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor microscales are not dissipative scales, but pass down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor microscales as a characteristic length scale and consider the energy cascade to contain only the largest and smallest scales; while the latter accommodate both the inertial subrange and the viscous sublayer. Nevertheless, Taylor microscales are often used in describing the term "turbulence" more conveniently as these Taylor microscales play a dominant role in energy and momentum transfer in the wavenumber space.

Although it is possible to find some particular solutions of the Navier–Stokes equations governing fluid motion, all such solutions are unstable to finite perturbations at large Reynolds numbers. Sensitive dependence on the initial and boundary conditions makes fluid flow irregular both in time and in space so that a statistical description is needed. The Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson) and the concept of self-similarity. As a result, the Kolmogorov microscales were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified.

A complete description of turbulence is one of the unsolved problems in physics. According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to Horace Lamb in a speech to the British Association for the Advancement of Science: "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather more optimistic."

Onset of turbulence

The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place

The onset of turbulence can be, to some extent, predicted by the Reynolds number, which is the ratio of inertial forces to viscous forces within a fluid which is subject to relative internal movement due to different fluid velocities, in what is known as a boundary layer in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation.

This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems, and is used to determine dynamic similitude between two different cases of fluid flow, such as between a model aircraft, and its full size version. Such scaling is not always linear and the application of Reynolds numbers to both situations allows scaling factors to be developed. A flow situation in which the kinetic energy is significantly absorbed due to the action of fluid molecular viscosity gives rise to a laminar flow regime. For this the dimensionless quantity the Reynolds number (Re) is used as a guide.

With respect to laminar and turbulent flow regimes:

• laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
• turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

The Reynolds number is defined as

${\displaystyle \mathrm{R}\mathrm{e}=\frac{\rho vL}{\mu },}$

where:

• ρ is the density of the fluid (SI units: kg/m³)
• v is a characteristic velocity of the fluid with respect to the object (m/s)
• L is a characteristic linear dimension (m)
• μ is the dynamic viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s)).

While there is no theorem directly relating the non-dimensional Reynolds number to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In Poiseuille flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 4000.

The transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.

Heat and momentum transfer

When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction coefficient.

Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual flow velocity v = (v_x,v_y) of every particle that passed through that point at any given time. Then one would find the actual flow velocity fluctuating about a mean value:

${\displaystyle v_x=\underset{\text{mean value}}{\underbrace{{\overline{v}}_x}}+\underset{\text{fluctuation}}{\underbrace{v_x^{\prime }}}\text{and}{v}_y={\overline{v}}_y+v_y^{\prime };}$

and similarly for temperature (T = T + T′) and pressure (P = P + P′), where the primed quantities denote fluctuations superposed to the mean. This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as a sub-field of fluid dynamics. While the mean values are taken as predictable variables determined by dynamics laws, the turbulent fluctuations are regarded as stochastic variables.

The heat flux and momentum transfer (represented by the shear stress τ) in the direction normal to the flow for a given time are

${\displaystyle \begin{array}{rl}q& =\underset{\text{experimental value}}{\underbrace{v_y^{\prime }\rho c_P{T}^{\prime }}}=-k_{\text{turb}}\frac{\mathrm{\partial }\overline{T}}{\mathrm{\partial }y};\\ \tau & =\underset{\text{experimental value}}{\underbrace{-\rho \overline{v_y^{\prime }{v}_x^{\prime }}}}={\mu }_{\text{turb}}\frac{\mathrm{\partial }{\overline{v}}_x}{\mathrm{\partial }y};\end{array}}$

where c_P is the heat capacity at constant pressure, ρ is the density of the fluid, μ_turb is the coefficient of turbulent viscosity and k_turb is the turbulent thermal conductivity.

Kolmogorov's theory of 1941

Richardson's notion of turbulence was that a turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by flow velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.

In his original theory of 1941, Kolmogorov postulated that for very high Reynolds numbers, the small-scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned). In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as L). Kolmogorov's idea was that in the Richardson's energy cascade this geometrical and directional information is lost, while the scale is reduced, so that the statistics of the small scales has a universal character: they are the same for all turbulent flows when the Reynolds number is sufficiently high.

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the kinematic viscosity ν and the rate of energy dissipation ε. With only these two parameters, the unique length that can be formed by dimensional analysis is

${\displaystyle \eta ={\left(\frac{{\nu }^3}{\epsilon }\right)}^{1/4}.}$

This is today known as the Kolmogorov length scale (see Kolmogorov microscales).

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length η, while the input of energy into the cascade comes from the decay of the large scales, of order L. These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length r) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. η ≪ r ≪ L). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called "inertial range").

Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range η ≪ r ≪ L are universally and uniquely determined by the scale r and the rate of energy dissipation ε.

The way in which the kinetic energy is distributed over the multiplicity of scales is a fundamental characterization of a turbulent flow. For homogeneous turbulence (i.e., statistically invariant under translations of the reference frame) this is usually done by means of the energy spectrum function E(k), where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u(x):

${\displaystyle \mathbf{u}(\mathbf{x})={\iiint }_{{\mathbb{R}}^3}\hat{\mathbf{u}}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}{\mathrm{d}}^3\mathbf{k},}$

where û(k) is the Fourier transform of the flow velocity field. Thus, E(k) dk represents the contribution to the kinetic energy from all the Fourier modes with k < |k| < k + dk, and therefore,

${\displaystyle \frac{1}{2}⟨u_i{u}_i⟩={\int }_0^{\mathrm{\infty }}E(k)\mathrm{d}k,}$

where 1/2⟨u_iu_i⟩ is the mean turbulent kinetic energy of the flow. The wavenumber k corresponding to length scale r is k = 2π/r. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

${\displaystyle E(k)=K_0{\epsilon }^{\frac{2}{3}}{k}^{-\frac{5}{3}},}$

where ${\displaystyle K_0\approx 1.5}$ would be a universal constant. This is one of the most famous results of Kolmogorov 1941 theory, and considerable experimental evidence has accumulated that supports it.

Outside of the inertial area, one can find the formula below :

${\displaystyle E(k)=K_0{\epsilon }^{\frac{2}{3}}{k}^{-\frac{5}{3}}\exp \left[-\frac{3K_0}{2}{\left(\frac{{\nu }^3{k}^4}{\epsilon }\right)}^{\frac{1}{3}}\right],}$

In spite of this success, Kolmogorov theory is at present under revision. This theory implicitly assumes that the turbulence is statistically self-similar at different scales. This essentially means that the statistics are scale-invariant and non-intermittent in the inertial range. A usual way of studying turbulent flow velocity fields is by means of flow velocity increments:

${\displaystyle \delta \mathbf{u}(r)=\mathbf{u}(\mathbf{x}+\mathbf{r})-\mathbf{u}(\mathbf{x});}$

that is, the difference in flow velocity between points separated by a vector r (since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of r). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation r when statistics are computed. The statistical scale-invariance without intermittency implies that the scaling of flow velocity increments should occur with a unique scaling exponent β, so that when r is scaled by a factor λ,

${\displaystyle \delta \mathbf{u}(\lambda r)}$

should have the same statistical distribution as

${\displaystyle {\lambda }^{\beta }\delta \mathbf{u}(r),}$

with β independent of the scale r. From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the flow velocity increments (known as structure functions in turbulence) should scale as

${\displaystyle ⟨(\delta \mathbf{u}(r){)}^n⟩=C_n⟨(\epsilon r{)}^{\frac{n}{3}}⟩,}$

where the brackets denote the statistical average, and the C_n would be universal constants.

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the n/3 value predicted by the theory, becoming a non-linear function of the order n of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov n/3 value is very small, which explain the success of Kolmogorov theory in regards to low order statistical moments. In particular, it can be shown that when the energy spectrum follows a power law

${\displaystyle E(k)\propto k^{-p},}$

with 1 < p < 3, the second order structure function has also a power law, with the form

${\displaystyle ⟨(\delta \mathbf{u}(r){)}^2⟩\propto r^{p-1},}$

Since the experimental values obtained for the second order structure function only deviate slightly from the 2/3 value predicted by Kolmogorov theory, the value for p is very near to 5/3 (differences are about 2%). Thus the "Kolmogorov −5/3 spectrum" is generally observed in turbulence. However, for high order structure functions, the difference with the Kolmogorov scaling is significant, and the breakdown of the statistical self-similarity is clear. This behavior, and the lack of universality of the C_n constants, are related with the phenomenon of intermittency in turbulence and can be related to the non-trivial scaling behavior of the dissipation rate averaged over scale r. This is an important area of research in this field, and a major goal of the modern theory of turbulence is to understand what is universal in the inertial range, and how to deduce intermittency properties from the Navier-Stokes equations, i.e. from first principles.