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Isotope fractionation in the hydrological cycle
Water is the primary source of hydrogen to all living organisms, so the isotopic composition of environmental water is a first-order control on that of the biosphere. The hydrological cycle moves water around different reservoirs on earth surface, during which hydrogen isotopes in water are significantly fractionated. As the primary moisture source to the atmosphere, the ocean has a relatively uniform water isotopic composition across the globe around 0‰ (VSMOW). Variations of δD larger than 10‰ in the ocean are generally confined to surface waters due to evaporation, sea ice formation, and addition of meteoric waters by precipitation, rivers or icebergs. In the hydrological cycle, the two major processes that fractionate hydrogen isotopes from the ocean water are evaporation and condensation. It should be pointed out that oxygen isotopic composition (18O/16O) of water is also an important tracer of the hydrological cycle, and cannot be separated from hydrogen isotopes when we talk about isotope fractionation processes in the hydrological cycle.

During evaporation of water from the ocean to the atmosphere, both equilibrium and kinetic isotope effects occur to determine the hydrogen and oxygen isotopic composition of the resulting water vapor. At the water-air interface, a stagnant boundary layer  is saturated with water vapor (100% relative humidity), and the isotopic composition of water vapor in the boundary layer reflects an equilibrium fractionation with liquid water. The liquid-vapor equilibrium fractionations for hydrogen and oxygen isotopes are temperature-dependent :

$$^2\varepsilon_{l-v}=52.612-76.248\times{1000 \over T(K)}+24.844\times{10^6 \over T(K)^2}$$ (‰)

$$^{18}\varepsilon_{l-v}=-2.0667-0.4156\times{1000 \over T(K)}+1.137\times{10^6 \over T(K)^2}$$ (‰)

The magnitude of this liquid-vapor equilibrium fractionation for hydrogen isotopes is approximately 8 times that of oxygen isotopes at earth surface temperatures, which reflects the relative mass differences of the two isotope systems (2H is 100% heavier than 1H, 18O is 12.5% heavier than 16O). Above the boundary layer, there is a transition zone with relative humidity less than 100%, and there is a kinetic isotope fractionation associated with water vapor diffusion from the boundary layer to the transition zone, which is empirically related to the relative humidity (h) :

$$^2\varepsilon_{bl-v}=12.5(1-h) $$ ‰

$$ ^{18}\varepsilon_{bl-v}=14.2(1-h)$$ ‰

The kinetic isotope effect associated with diffusion reflects the mass difference of the heavy-isotope substituted water molecules (HD16O and H218O) relative to the normal isotopologue (H216O).

After water is evaporated to the atmosphere, it is transported and returned to earth surface through condensation and precipitation. Condensation of water vapor occurs in ascending air masses which develops a lower temperature and saturation vapor pressure. Since the cooling and condensation happens at relatively slow rates, it is a process with equilibrium isotope effects. However, as water vapor is progressively condensed and lost from the air during moisture transport, the isotopic composition of the remaining vapor, as well as the resulting precipitation, can be largely depleted due to the process of Rayleigh distillation. The equation for Rayleigh distillation is : $$R_r/R_0=f^{\alpha-1} $$

In the equation, R0 represents the isotope ratio in the initial water vapor, Rr represents the isotope ratio in the remaining water vapor after some condensation, f is the fraction of water vapor remaining in the air, and α is the liquid-vapor equilibrium fractionation factor (α=1+ε). The isotopic composition of the resulting precipitation (Rp) can be derived from the composition of the remaining vapor:

$$R_p/R_r=(1+\delta_p)/(1+\delta_r)=\alpha=1+\varepsilon $$

As f decreases progressively during condensation, the remaining vapor becomes more and more depleted in the heavy isotopes, and the magnitude of depletion becomes larger as f approaches zero. The Rayleigh distillation process can explain some first-order spatial patterns observed in the isotopic composition of precipitation across the globe, including isotopic depletion from the tropics to the poles, isotopic depletion from coastal to inland regions, and isotopic depletion with elevation over a mountain range, all of which are associated with progressive moisture loss during transport. The Rayleigh distillation model can also be used to explain the strong correlation between δD and δ18O observed in global precipitation, expressed as the global meteoric water line (GMWL): δD=8δ18O+10 (later updated to δD=8.17±0.07 δ18O+11.27±0.65 ) The slope of the GMWL reflects the relative magnitude of hydrogen and oxygen isotope fractionation during condensation. It should be noted that the intercept of GMWL is non-zero (called deuterium-excess, or d-excess), which means ocean water does fall on GMWL. This is associated with the kinetic isotope effect during evaporation when water vapor diffuses from the saturated boundary layer to the unsaturated transition zone, and cannot by explained by the Rayleigh model. Nevertheless, the robust pattern in GMWL strongly suggests a single dominant moisture source to the global atmosphere, which is the tropical western Pacific. It should also be pointed out that a local meteoric water line can have a different slope and intercept from the GMWL, due to differences in humidity and evaporation intensity at different places. Hydrogen and oxygen isotopes in water thus serve as an excellent tracer of the hydrological cycle both globally and locally.

Applications
Based on the processes that fractionate isotopes in the hydrological cycle, isotopic composition of meteoric water can be used to infer related environmental variables such as air temperature, precipitation amount, past elevations, lake levels, as well as to trace moisture sources. These studies form the field of isotope hydrology. Examples of isotope hydrology applications include the following:

Temperature reconstruction
Isotopic composition of precipitation can be used to infer changes in air temperature based on the Rayleigh process. Lower temperature corresponds to lower saturation vapor pressure, which means more condensation has happened to drive the residual vapor isotopically depleted. The resulting precipitation thus has a more negative δD and δ18O value at lower temperature. This precipitation isotope thermometer is most sensitive at low temperatures, and widely applied at high latitudes. For example, δD and δ18O were found to have a temperature sensitivity of 8‰/°C and 0.9‰/°C in Antarctic snow, and a sensitivity of 5.6‰/°C and 0.69‰/°C across Arctic sites. δD and δ18O of ice cores in Greenland, Antarctica and alpine glaciers are important archives of temperature change in the geological past.

Precipitation amount effect
In contrast to temperature control at high latitudes, the isotopic composition of precipitation in mainly influenced by rainfall amount (negative correlation) in the tropics, and the "amount effect" is also observed for summer precipitation in the subtropics. Willi Dansgaard, who first proposed the term "amount effect", suggested several possible reasons for the correlation: (1) As the cooling and condensation progress, the rainfall isotopic composition reflects an integrated isotopic depletion by the Rayleigh process; (2) Small amount of rainfall is more likely to be influenced by evaporation and exchange with surrounding moisture, which tend to make it more isotopically enriched. At low latitudes, the amount effect for δ18O is around -1.6‰ per 100 mm precipitation increase at island stations, and -2.0‰ per 100 mm at continental stations. It was also noted that the amount effect was most pronounced when comparing isotopic composition of monthly precipitation at different places in the tropics. The amount effect is also expected for hydrogen isotopes, but there are not as many calibration studies. Across southeast Asia, the δD sensitivity to monthly precipitation amount varies between -15 and -25‰/100mm depending on location. In temperate regions, the isotopic composition of precipitation in dominated by rainfall amount in summer months, but more controlled by temperature in the winter. The amount effect may also be complicated by changes in regional moisture sources. Reconstructions of rainfall amount in the tropics in the geological past are mostly based on δ18O of speleothems or δD of biogenic lipids , both of which are thought of as proxies for the isotopic composition of precipitation.

Isotope Hydrology
Hydrogen and oxygen isotopes also work as a tracer for water budget in terrestrial reservoirs, including lakes, rivers, groundwater and soil water. For a lake, both the amount of water in the lake and the isotopic composition of the water are determined by a balance between inputs (precipitation, stream and ground water inflow) and outputs (evaporation, stream and ground water outflow). The isotopic composition of lake water can often be used to track evaporation, which causes isotope enrichment in the lake water, as well as a δD-δ18O slope that is shallower than the meteoric water line. The isotopic composition of river water is highly variable and have complicated sources over different timescales, but can generally be treated as a two-endmember mixing problem, a base-flow endmember (mainly ground water recharge) and an overland-flow endmember (mainly storm events). The isotope data suggest that the long-term integrated base-flow endmember is more important in most rivers, even during peak flows in summer. Systematic river isotope data were collected across the world by the Global Network of Isotopes in Rivers (GNIR).The isotopic composition of groundwater can also be used to trace its sources and flow paths. An example is a groundwater isotope mapping study in Sacramento, California, which showed lateral flow of river water with a distinct isotope composition into the groundwater that developed a significant water table depression due to pumping for human use. The same study also showed an isotopic signal of agricultural water being recharged into the giant alluvial aquifer in Central Valley, California. Finally, the isotopic composition of soil water is important for study of plants. Below the water table, the soil has a relatively constant source of water with a certain isotopic composition. Above the water table, the isotopic composition of soil water is enriched by evaporation until a maximum at the surface. The vertical profile of isotopic composition of soil water is maintained by the diffusion of both liquid and vapor water. A comparison of soil water and plant xylem water δD can be used to infer the depth at which plant roots get water from the soil.

Ice Core Records
Isotopic composition in ice core s from continental ice sheets and alpine glaciers have been developed as temperature proxies since the 1950's. Samuel Epstein was one of the first to demonstrate the applicability of this proxy by measuring oxygen isotopes in Antarctic snow, and also pointed out complications in the stable isotope-temperature correlation brought by the history of air masses from which the snow formed. Ice cores in Greenland and Antarctica can be thousands of meters thick and record snow isotopic composition of the past few glacial-interglacial cycles. Ice cores can be dated by layer counting on the top and ice flow modeling at depth, with additional age constraints by volcanic ash. Cores from Greenland and Antarctica can be aligned in age at high-resolution by comparing globally well-mixed trace gas (e.g. CH4) concentrations in the air bubbles trapped in cores. Some of the first ice core records from Greenland and Antarctica with age estimates go back to the last 100,000 years, and showed a depletion in δD and δ18O in the last ice age. The ice core record has since been extended to the last 800,000 years in Antarctica, and at least 250,000 years in Greenland. One of the best δD-based ice core temperature record is from Vostok ice core in Antarctica, which goes back to 420,000 years. The δD-temperature (of the inversion layer where snow forms) conversion in east Antarctica based on modern spatial gradient of δD (9‰/°C) is ΔTI=(ΔδDice-8Δδ18Osw)/9, which takes into account variations in seawater isotopic composition caused by global ice volume changes. The Vostok ice core record shows some very important results: (1) A consistent δD depletion of ~70‰ during the last four glacial periods compared to interglacial times, corresponding to a cooling of 8°C in Antarctica; (2) A consistent drop of atmospheric CO2 concentration of 100 ppmv and CH4 drop of ~300 ppbv during glacial times from interglacials, suggesting role of greenhouse gases in regulating global climate; (3) Antarctic air temperature and greenhouse gas concentration changes lead global ice volume and Greenland air temperature changes during glacial terminations, and greenhouse gases may be an amplifier of insolation forcing during glacial-interglacial cycles. Greenland ice core isotope records, in addition to showing glacial-interglacial cycles, also shows millennial-scale climate oscillations that may reflect reorganization in ocean circulation caused by ice melt charges. There have also been ice core records generated in alpine glacials on different continents. A record from Andes Mountains in Peru shows a temperature decrease of 5-6°C in the tropics during the last ice age. A record from the Tibetan plateau shows a similar isotope shift and cooling during the last ice age. Other existing alpine glacial isotope records include Mount Kilimanjaro in Tanzania, Mount Altai and West Belukha Plateau in Russia, Mount Logan in Canada, Fremont Glacier in Wyoming, USA, and Illimani Ice Core in Bolivia, most of which cover an interval of the Holocene epoch.

Stable Isotope Paleoaltimetry
The possibility of using water isotope depletion with elevation to reconstruct paleoaltimetry has been demonstrated as early as the late 1960's, when Caltech geochemist Samuel Epstein tried to collect rainwater at different elevations in a single storm. The δ18O and δD lapse rates vary within -1 to -5‰/km and -10 to -40‰/km respectively, but can vary with locations and seasons, and are not exactly linear with altitude. One of the first studies in stable isotope paleoaltimetry demonstrated a meteoric water δD signature of -90 to -139‰ in fluid inclusions in quartz and adularia in an epithermal gold-silver deposit in Nevada, and suggested the applicability of stable isotopes in reconstruction of ancient topography in the Great Basin. The hydrogen and oxygen isotopes of hydrous silicate minerals have since then been used to reconstruct topographic histories in mountain ranges across the world, including the North American Cordillera, the Rocky Mountains, the Himalayas, the European Alps, and Southern Alps in New Zealand. Lab experiments with clay minerals have shown that the hydrogen and oxygen isotope compositions are relatively resistant to alteration at moderate temperature (<100°C), and can preserve the original meteoric water signal. One important effect of mountain ranges on rainfall stable isotopes is the rain shadow effect, in which an isotopic depletion happens on precipitation on the leeward side compared to the windward side. A change on the difference in isotopic composition of precipitation on two sides of a mountain can be used to infer the magnitude of the rain shadow effect. In one such study, an isotope enrichment was observed in smectite on the east side of the Sierra Nevada in California from mid-Miocene to late Pliocene, suggesting a decrease in elevation during this period. Another study found δD values around -140‰ in muscovite in the North America Cordillera during the early Eocene, which would suggest an elevation 1000m higher than today at the time. In addition to hydrous minerals, hydrogen isotopes in biomarkers such as leaf waxes have also been developed for paleoaltimetry studies. The δD lapse rate in leaf waxes (-21‰/km) falls in the range of meteoric water observations. As an example study, leaf wax δD data have been used to confirm hydrous mineral paleoaltimetry for the high elevation of the Sierra Nevada during the Eocene.

Cellulose Records
The carbon-bonded hydrogen isotopic composition of cellulose, as inherited from leaf water, has the potential of preserving the original meteoric water signal. This was first demonstrated in the 1970's. In a systematic survey across North America, tree cellulose δD was found to have a temperature sensitivity of 5.8‰/°C, similar to precipitation δD sensitivity of 5.6‰/°C. This spatial correlation may be complicated by local effects of soil evaporation and leaf transpiration, and the spatial gradient may not be representative of temporal changes in tree ring cellulose at a single place. The mechanism that generates the δD signal in cellulose from meteoric water is not completely understood, but at least include leaf water transpiration, photosynthesis of carbohydrates, synthesis of cellulose from photosynthetic sugars, and exchange of sugars with xylem water. Modeling studies show that observed tree ring cellulose δD can be produced when 36% of the hydrogen in sugars can exchange with xylem water, and effects such as humidity and rainfall seasonality may complicate the cellulose δD proxy. Despite these complications, tree ring δD have been used for paleoclimate reconstructions of the past few millennia. For example, a tree ring cellulose δD records from pine trees in the White Mountains, California shows a 50‰ depletion from 6800 year ago to present. The cooling trend since the mid-Holocene thermal maximum is consistent with ice core and pollen records, but the corresponding magnitude of cooling is elusive due to complicated influences from local effects such as humidity and soil water composition. The meaning of isotopes in cellulose and its applications is still an area of active study.