Virtual temperature
Virtual temperature is a fictitious temperature in atmospheric thermodynamics that represents the temperature a sample of dry air would need to achieve the same density as a given sample of moist air at the same total pressure and moisture content.[1] It accounts for the lower molecular weight of water vapor compared to dry air, making moist air less dense than dry air at the same actual temperature and pressure, and thus virtual temperature is always slightly higher than the actual temperature for humid conditions.[2] The concept derives from the ideal gas law applied to moist air, where total pressure p = p_d + e (dry air partial pressure plus water vapor partial pressure) and total density \rho = \rho_d + \rho_v + \rho_l (dry air, vapor, and liquid water densities), leading to the defining relation p = \rho R_d T_v, with R_d as the gas constant for dry air.[2] A common approximation for the virtual temperature T_v is T_v \approx T (1 + 0.608 q), where T is the actual temperature in Kelvin and q is the water vapor mixing ratio (mass of vapor per mass of dry air); this factor 0.608 arises from the ratio of gas constants for water vapor and dry air (\epsilon \approx 0.622).[1] For more precision including liquid water, the formula extends to T_v = T \frac{1 + q/\epsilon}{1 + q + l}, where l is the liquid water mixing ratio.[2] In atmospheric science, virtual temperature is essential for evaluating air parcel buoyancy and static stability, as it allows the use of dry-air equations for moist conditions in calculations like convective available potential energy (CAPE), which is critical for forecasting thunderstorm development.[3] It also facilitates remote sensing of atmospheric profiles, such as via radio acoustic sounding systems (RASS), and is used in modeling turbulent fluxes and hydrostatic balance in weather prediction models.[3] By simplifying the treatment of moisture's density effects, virtual temperature enhances accuracy in cloud physics, numerical weather prediction, and climate simulations without altering the fundamental dry-air framework.[1]Overview
Definition
Virtual temperature, denoted T_v, is defined as the temperature that a theoretical parcel of dry air would need to possess in order to have the same total pressure and density as an actual parcel of moist air at the same location.[4][5] This construct arises in atmospheric thermodynamics to account for the effects of water vapor on air density while allowing moist air behavior to be approximated using dry air equations.[4] The definition presupposes the ideal gas law for dry air, expressed as p = \rho R_d T, where p is the total pressure, \rho is the density, R_d is the specific gas constant for dry air (approximately 287 J kg⁻¹ K⁻¹), and T is the temperature. In this framework, T_v adjusts the temperature term to reflect the lower density of moist air due to the lighter molecular weight of water vapor compared to dry air constituents. In unsaturated moist air, T_v exceeds the actual temperature T because the presence of water vapor reduces the parcel's density relative to dry air at the same T and p.[6] Conversely, in saturated air with suspended liquid water droplets (known as liquid water loading), the added mass of these denser droplets increases the overall density, resulting in T_v being lower than T.[7] Virtual temperature is measured in kelvin and functions as a scalar multiplier that can substitute for actual temperature in thermodynamic relations originally formulated for dry air.Significance
The virtual temperature serves a critical purpose in atmospheric science by permitting the dry-air equation of state to be applied to moist air through a simple temperature scaling, which streamlines hydrostatic balance computations and thermodynamic analyses in numerical weather prediction and climate models. This adjustment accounts for the lower density of water vapor compared to dry air, avoiding the need for separate moist-air formulations and reducing computational complexity in regions with significant humidity.[4] The concept of virtual temperature was first introduced by Cato Guldberg and Henrik Mohn in 1876.[8] It has since been used to overcome shortcomings in representations of moist air dynamics, as exemplified in D. K. Lilly's 1968 analysis of cloud-topped mixed layers, where virtual temperature facilitated more accurate modeling of buoyancy and stability under strong inversions.[9] Without incorporating virtual temperature, density miscalculations in humid environments can lead to errors in pressure-to-height conversions of up to 10-20 meters across tropospheric layers, particularly in the tropics where moisture content is high.[10] A key impact of virtual temperature lies in its representation of the vapor buoyancy effect, which introduces an approximate 1 K warming in the tropical troposphere relative to actual temperature; this enhancement promotes greater buoyancy in moist air parcels, thereby intensifying convective processes and contributing to increased clear-sky outgoing longwave radiation by about 1 W m⁻² globally. Such effects help stabilize the tropical climate through negative feedbacks that mitigate excessive warming in dry subsiding regions.[11][12]Physical Principles
Air Density and Water Vapor Effects
The density of moist air arises from its composition as a mixture of dry air and water vapor, treated as ideal gases following Dalton's law of partial pressures. According to this law, the total atmospheric pressure p equals the sum of the partial pressure of dry air p_d and the vapor pressure e: p = p_d + e. This partial pressure framework allows the densities of each component to be calculated separately using the ideal gas law.[13][14] The molecular weight of dry air is approximately 29 g/mol, primarily from nitrogen (28 g/mol) and oxygen (32 g/mol), whereas water vapor has a lower molecular weight of 18 g/mol. The total density of moist air \rho is thus the sum of the dry air density \rho_d and the water vapor density \rho_v: \rho = \rho_d + \rho_v. Since water vapor molecules are lighter than the average dry air molecules they displace, the presence of water vapor reduces the overall mass per unit volume, making moist air less dense than dry air at the same temperature and pressure.[13][15][16] This density reduction stems from variations in the ideal gas law for moist air. The specific gas constant for dry air is R_d \approx 287 J kg⁻¹ K⁻¹, while for water vapor it is R_v \approx 461 J kg⁻¹ K⁻¹, reflecting the inverse relationship with molecular weight. The effective gas constant R for moist air can be expressed as R = R_d \left[1 - \frac{e}{p} (1 - \epsilon)\right]^{-1}, where \epsilon = 0.622 is the ratio of the molecular weight of water vapor to dry air (\epsilon = M_v / M_d). At constant temperature T and total pressure p, the higher effective R causes moist air to occupy a greater volume than dry air, further lowering its density by replacing heavier dry air molecules with lighter water vapor ones.[13][17][14] In humid regions like the tropics, where mixing ratios can exceed 20 g kg⁻¹, this effect becomes notable; for instance, at 30°C and a mixing ratio of 20 g kg⁻¹, the density of moist air is reduced by approximately 1.2% compared to dry air at the same conditions. Overall, density reductions reach up to 2-3% in highly saturated tropical air, enhancing buoyancy and influencing atmospheric processes. This physical mechanism underpins the virtual temperature concept, which equates the density of moist air to that of dry air at an adjusted temperature.[13][17][15]Vapor Buoyancy Mechanism
In the atmosphere, the buoyancy acceleration of a rising air parcel is determined by b = g \frac{\Delta \rho}{\rho}, where g is the acceleration due to gravity, \Delta \rho is the density difference between the parcel and its environment, and \rho is the environmental density. This acceleration drives vertical motion, with positive buoyancy occurring when the parcel density is lower than the surrounding air. Water vapor contributes to this by reducing the overall density of the moist parcel relative to dry air at the same temperature and pressure, as its molecular weight (18 g/mol) is lower than that of dry air (29 g/mol), thereby enhancing the ascent rate compared to an equivalent dry parcel.[18][19] The vapor buoyancy effect, quantified through virtual temperature, provides an equivalent temperature perturbation that amplifies buoyancy in moist air. For a water vapor mixing ratio of 10 g/kg, this effect corresponds to roughly 1-2 K of warming in typical tropospheric conditions (around 280-300 K), promoting stronger updrafts in moist convective processes by making vapor-laden parcels effectively warmer and lighter. This dynamic enhancement is particularly pronounced in regions with high humidity, where even modest increases in mixing ratio can significantly boost convective vigor.[11][13] A key aspect of this mechanism is its dependence on the phase of water in the parcel. During unsaturated ascent, the virtual temperature directly increases buoyancy owing to the low density of water vapor alone. However, once saturation occurs and condensation forms clouds, the added mass of liquid water droplets (liquid loading) increases the parcel's density, often counteracting or reversing the vapor-induced buoyancy gain and potentially slowing or inhibiting further ascent.[20][21] This mechanism also influences Earth's global energy budget, particularly in the tropics. By elevating the virtual temperature in moist regions and inducing compensatory warming in adjacent drier columns to maintain hydrostatic balance, the vapor buoyancy effect enhances clear-sky outgoing longwave radiation by approximately 1-3 W/m², providing a stabilizing feedback that increases radiative cooling as surface temperatures rise.[11]Formulation
Derivation
The derivation of the virtual temperature begins with the equation of state for moist air, treated as an ideal gas mixture of dry air and water vapor under the assumptions of thermodynamic equilibrium and Dalton's law of partial pressures. The partial pressure of dry air is p_d = p - e, where p is the total pressure and e is the water vapor pressure.[22] The density of dry air is given by \rho_d = \frac{p_d}{R_d T} = \frac{p - e}{R_d T}, where R_d is the specific gas constant for dry air and T is the actual temperature in Kelvin. Similarly, the density of water vapor is \rho_v = \frac{e}{R_v T}, with R_v as the specific gas constant for water vapor. The total density \rho of unsaturated moist air (neglecting liquid water) is then \rho = \rho_d + \rho_v = \frac{p - e}{R_d T} + \frac{e}{R_v T}.[22] The virtual temperature T_v is defined such that the moist air density equals that of dry air at the same total pressure p and temperature T_v, using the dry air gas constant: \rho = \frac{p}{R_d T_v}.[22] Equating the two expressions for \rho, \frac{p}{R_d T_v} = \frac{p - e}{R_d T} + \frac{e}{R_v T}. Multiplying through by R_d T yields \frac{p T}{T_v} = (p - e) + e \frac{R_d}{R_v}. The ratio of gas constants is \varepsilon = \frac{R_d}{R_v} \approx 0.622, derived from the molar masses of dry air (M_d \approx 28.97 g/mol) and water vapor (M_v = 18 g/mol) via \varepsilon = \frac{M_v}{M_d}, since R_d = \frac{R^*}{M_d} and R_v = \frac{R^*}{M_v} with R^* the universal gas constant. Substituting gives \frac{p}{T_v} = \frac{p - e(1 - \varepsilon)}{T}, and solving for T_v produces the exact formula T_v = \frac{T}{1 - \frac{e}{p} (1 - \varepsilon)}.[22] This derivation assumes ideal gas behavior for both components, negligible liquid water content (valid for unsaturated air), and that all constituents share the same temperature T. It is applicable below the homopause (approximately 90 km altitude) where dry air is well-mixed.[22]Approximate Expressions
In atmospheric science, linear approximations for virtual temperature simplify computations by relating it directly to the actual air temperature and a measure of humidity, bypassing the need for precise vapor pressure calculations. A standard form uses specific humidity q (in kg/kg):T_v \approx T (1 + 0.608 q),
valid for q < 0.02 kg/kg, which encompasses most tropospheric scenarios where moisture content is moderate.[23] This expression assumes small perturbations from dry air conditions and leverages the ideal gas law for moist air.[16] The coefficient 0.608 arises from the physical properties of air components, specifically (1 - \epsilon)/\epsilon \approx 0.608, where \epsilon = 0.622 is the ratio of the molecular weight of water vapor to dry air (or equivalently, the ratio of their specific gas constants). This approximation holds under the condition of small e/p, the ratio of vapor pressure to total pressure.[13] For cases where mixing ratio w (in g/kg) is the available humidity metric, equivalent approximations are:
T_v \approx T + \frac{w}{6}
(with T and T_v in °C), or in Kelvin,
T_v \approx T \left(1 + 0.608 \frac{w}{1000}\right).
These derive from the specific humidity form by noting w \approx 1000 q for low moisture levels, facilitating practical use in field measurements or models.[24] These linear forms yield errors below 0.5 K in typical tropospheric conditions (e.g., w < 20 g/kg), as higher-order terms in the expansion are negligible; however, inaccuracies grow in extreme humidity exceeding these limits.[24] They prove valuable for rapid assessments in radiosonde analyses or forecasting tools lacking full thermodynamic data.[13]