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Thermal wind

The thermal wind is a fundamental concept in meteorology representing the vertical shear of the geostrophic wind caused by horizontal temperature gradients in the atmosphere. It is defined as the vector difference between the geostrophic wind at an upper pressure level and that at a lower pressure level, such that \mathbf{V}_T = \mathbf{V}_g(\text{upper}) - \mathbf{V}_g(\text{lower}), and it arises in baroclinic atmospheres where temperature variations with height lead to changes in wind speed and direction.^[1]^[2]^[3] This shear is not an actual wind but a diagnostic tool that links thermal structure to atmospheric circulation, with the thermal wind vector pointing parallel to isotherms and having cold air to its left in the Northern Hemisphere.^[1]^[2] The thermal wind relation is derived from the geostrophic balance equations, v_g = \frac{1}{f} \frac{\partial \Phi}{\partial x} and u_g = -\frac{1}{f} \frac{\partial \Phi}{\partial y}, combined with the hydrostatic equation \frac{\partial \Phi}{\partial p} = -\frac{R T}{p} and the ideal gas law, yielding \mathbf{V}_T = \frac{R}{f} \hat{k} \times \nabla_p T \log\left(\frac{p_1}{p_2}\right), where f is the Coriolis parameter, R is the gas constant for dry air, and \nabla_p T is the horizontal temperature gradient on a pressure surface.^[1]^[2]^[3] This formulation shows that the magnitude and direction of the wind shear are directly proportional to the temperature gradient, with stronger gradients producing greater shear; for instance, the x-component is u_T = -\frac{R}{f} \left\langle \frac{\partial T}{\partial y} \right\rangle \log\left(\frac{p_1}{p_2}\right) and the y-component is v_T = \frac{R}{f} \left\langle \frac{\partial T}{\partial x} \right\rangle \log\left(\frac{p_1}{p_2}\right).^[2]^[3] In midlatitudes, the thermal wind explains the predominantly westerly (eastward) flow of surface and upper-level winds, as the poleward decrease in temperature creates an equatorward-pointing temperature gradient that, under geostrophic balance, results in a shear reinforcing westerly winds with height, contributing to phenomena like the jet stream.^[1]^[3] It is crucial for weather forecasting, as it ensures consistency between observed wind, geopotential height, and temperature fields, and helps predict temperature advection—warm advection causes winds to veer with height, while cold advection causes backing.^[1]^[2] The concept applies primarily to large-scale, synoptic flows where geostrophic and hydrostatic approximations hold, contrasting with barotropic conditions lacking such shear.^[1]^[3]

Conceptual Foundation

Definition

The thermal wind is defined as the vector difference between the geostrophic winds at two different pressure levels in the atmosphere, resulting exclusively from horizontal temperature gradients in a rotating fluid such as Earth's atmosphere.^[4] This difference represents the vertical shear of the geostrophic wind, which is parallel to the isotherms and to lines of constant thickness between the pressure levels, with its magnitude proportional to the strength of the horizontal temperature contrast.^[5] In essence, the thermal wind quantifies how temperature variations across horizontal distances induce changes in wind speed and direction with height, a key feature in baroclinic conditions where isobaric and isosteric surfaces are not coincident.^[6] The geostrophic wind itself is the hypothetical wind that arises from a balance between the Coriolis force acting on moving air and the horizontal pressure gradient force, approximating large-scale atmospheric flow in the absence of friction. Horizontal temperature gradients, expressed mathematically as ∂T/∂x or ∂T/∂y, reflect spatial variations in temperature that alter air density and thus pressure at different heights, leading to the thermal wind shear.^[7] These gradients play a central role in baroclinic atmospheres, where horizontal temperature differences prevent pressure surfaces from being level, fostering vertical variations in geostrophic flow that the thermal wind captures.^[8] The term "thermal wind" was proposed in 1918 by British meteorologist Ernest Gold, emerging from early 20th-century advancements in synoptic meteorology that integrated temperature observations with upper-air wind patterns.^[9] It provided a diagnostic tool to relate surface weather maps to upper-level dynamics, influencing the development of numerical weather prediction. The thermal wind is typically expressed in units of meters per second (m/s) for the total vector difference between levels, though it is often conceptualized as shear in m/s per kilometer of height; in mid-latitudes, magnitudes commonly range from 10 to 50 m/s over the tropospheric depth, reflecting strong meridional temperature contrasts.^[10]

Physical Mechanism

The thermal wind arises from the buoyancy-driven response of the atmosphere to horizontal temperature gradients. In regions where air is warmer, such as toward the equator compared to the poles, the less dense warm air expands vertically, resulting in a slower decrease of pressure with height within those columns. This expansion causes pressure surfaces to rise higher over warm areas than over adjacent cold regions, where denser air leads to a more rapid pressure drop. Consequently, at upper levels, the horizontal pressure gradient becomes steeper over the warm side, enhancing the pressure gradient force aloft and driving stronger geostrophic winds at higher altitudes.^[11]^[12] Earth's rotation plays a crucial role through the Coriolis effect, which deflects the response to these pressure differences. As the enhanced pressure gradient aloft accelerates air masses, the Coriolis force turns the flow perpendicular to the gradient, aligning geostrophic winds parallel to the isotherms—the lines of constant temperature. This deflection ensures that the vertical shear in the wind, known as the thermal wind, points perpendicular to the temperature gradient (parallel to the isotherms), with cold air typically to the left in the [Northern Hemisphere](/page/Northern Hemisphere). The result is a systematic increase in wind speed with height, particularly in the westerly direction in mid-latitudes, without altering the wind direction at the surface.^[11]^[13] This phenomenon is fundamentally tied to baroclinicity, the condition where isotherms and isobars are not aligned, allowing temperature variations on constant-pressure surfaces. In baroclinic atmospheres, prevalent in mid-latitudes, these misalignments produce the vertical wind shear characteristic of the thermal wind. By contrast, in barotropic conditions—common in the deep tropics—temperature surfaces parallel pressure surfaces, resulting in uniform density with height and no thermal wind shear, as geostrophic winds remain constant across levels.^[12]^[13] Qualitatively, this can be visualized through sketches of sloped isobaric surfaces: over a warm region, isobars tilt upward with height, creating a funnel-like pressure gradient that amplifies from surface to upper troposphere, while wind vectors depict a progressive strengthening and alignment parallel to the underlying isotherms. Such diagrams illustrate how the initial surface pressure gradient, often modest due to balanced surface temperatures, evolves into pronounced upper-level gradients solely from thermal expansion effects.^[11]^[13]

Theoretical Framework

Geostrophic Balance Prerequisites

The hydrostatic approximation forms a fundamental prerequisite for deriving the thermal wind relation, positing that vertical accelerations in the atmosphere are negligible compared to gravitational acceleration, thereby simplifying the vertical momentum equation. This assumption yields the hydrostatic balance equation, expressed as \frac{\partial p}{\partial z} = -\rho [g](/page/G), where p is pressure, z is height, \rho is air density, and g is gravitational acceleration. The approximation holds because typical vertical velocities in large-scale atmospheric flows are on the order of centimeters per second, resulting in accelerations far smaller than g \approx 9.8 \, \mathrm{m/s^2}, allowing pressure to decrease with height primarily due to the weight of the overlying air column.^[14] Complementing the hydrostatic approximation, geostrophic balance assumes that, for large-scale horizontal flows, the Coriolis force counteracts the pressure gradient force, neglecting local accelerations, friction, and curvature effects. This leads to the geostrophic wind equation: f \mathbf{k} \times \mathbf{V}_g = -\frac{1}{\rho} \nabla p, where f = 2 \Omega \sin \phi is the Coriolis parameter (\Omega is Earth's angular velocity and \phi is latitude), \mathbf{V}_g is the geostrophic velocity, and \nabla p is the horizontal pressure gradient. Under steady-state conditions, this balance implies that winds flow parallel to isobars, with speed inversely proportional to f and directly to the pressure gradient magnitude.^[1] These approximations are valid primarily for synoptic-scale motions in mid-latitudes, where horizontal scales exceed approximately 100 km, ensuring a small Rossby number (Ro = U / f L < 0.1) that justifies neglecting accelerations relative to Coriolis forces, and where f \neq 0 to avoid equatorial singularities. They break down near the equator due to vanishing f, in small-scale turbulent flows (length scales <10 km) where friction dominates, or during rapid transients like convective storms where accelerations become significant. Hydrostatic validity similarly requires horizontal scales much larger than the atmospheric depth scale (~10 km) to minimize vertical motion effects.^[15]^[16] The thermal wind relation is derived using the hydrostatic and geostrophic approximations. These approximations are central to the scale analysis that justifies the primitive equations, which govern large-scale atmospheric motion and include the hydrostatic balance, continuity, and thermodynamic equations while retaining time-dependent terms for prognostic evolution. These equations, derived from scale analysis of large-scale flows, provide the balanced state from which vertical wind shear due to horizontal temperature gradients—the essence of thermal wind—can be diagnosed without resolving smaller-scale dynamics. Seminal developments in primitive equation models, such as those used in global circulation simulations, underscore their role in capturing baroclinic instabilities central to thermal wind phenomena.^[17]^[18]

Derivation and Equations

The derivation of the thermal wind relation begins with the geostrophic balance equations in pressure coordinates, where the horizontal pressure gradient force is balanced by the Coriolis force.^[19] The geostrophic wind components are given by

u_g = -\frac{1}{f} \left( \frac{\partial \Phi}{\partial y} \right)_p, \quad v_g = \frac{1}{f} \left( \frac{\partial \Phi}{\partial x} \right)_p,

where \Phi is the geopotential height, f = 2 \Omega \sin \phi is the Coriolis parameter (assumed constant under the f-plane approximation), and subscripts denote evaluation on constant-pressure surfaces.^[19] To obtain the vertical shear, differentiate these equations with respect to \ln p, using the hydrostatic relation \frac{\partial \Phi}{\partial p} = -\frac{R T}{p}, where R is the gas constant for dry air and the ideal gas law p = \rho R T holds.^[19] Interchanging the order of differentiation (valid under horizontal incompressibility, assuming no significant horizontal variations in density affect the vertical derivatives) yields the scalar components of the thermal wind shear:

\frac{\partial u_g}{\partial \ln p} = \frac{R}{f} \left( \frac{\partial T}{\partial y} \right)_p, \quad \frac{\partial v_g}{\partial \ln p} = -\frac{R}{f} \left( \frac{\partial T}{\partial x} \right)_p.

^[19] In vector form, this is

\frac{\partial \vec{V}_g}{\partial \ln p} = \frac{R}{f} \hat{k} \times \nabla_p T,

where \vec{V}_g = (u_g, v_g) and \nabla_p is the horizontal gradient on pressure surfaces; note that \ln p increases downward, so the shear direction reflects changes from lower to upper levels.^[19] For the integrated form between two pressure levels p_1 (lower) and p_2 (upper, p_1 > p_2), subtract the geostrophic winds at these levels:

\vec{V}_T = \vec{V}_{g2} - \vec{V}_{g1} = \frac{R \ln(p_1 / p_2)}{f} \hat{k} \times \nabla_p \bar{T},

where \bar{T} is the mean temperature in the layer, derived by integrating the differential form assuming constant gradients.^[20] In height coordinates, the equivalent integrated thermal wind uses the hypsometric relation for layer thickness \Delta z = \frac{R \bar{T}}{g} \ln(p_1 / p_2), leading to

\vec{V}_T = \frac{g}{f} \hat{k} \times \nabla_h \Delta z = \frac{R \ln(p_1 / p_2)}{f} \hat{k} \times \nabla_h \bar{T},

where \nabla_h is the horizontal gradient and g is gravitational acceleration; the differential shear is \frac{\partial \vec{V}_g}{\partial z} = \frac{g}{f T} \hat{k} \times \nabla_h T under approximations of small temperature variations.^[21]^[20] The vector form indicates that the thermal wind is perpendicular to the horizontal temperature gradient \nabla T and parallel to contours of constant thickness (which approximate isotherms for mean layer temperature).^[19]

Meteorological Applications

Jet Stream Dynamics

Jet streams represent narrow bands of strong westerly winds in the upper troposphere, primarily driven by the thermal wind relation arising from meridional temperature gradients. In mid-latitudes, particularly along the polar front where cold polar air meets warmer mid-latitude air, these gradients create significant baroclinicity, leading to vertical wind shear through thermal wind effects. This shear typically ranges from 5 to 10 m/s per km in the troposphere, resulting in total increases of 20-50 m/s over altitudes of 5-10 km, which aligns with the formation of the polar jet around 50-60° latitude and the subtropical jet near 30° latitude in both hemispheres.^[22]^[23]^[24] The vertical structure of jet streams manifests as a maximum in the thermal wind profile, where geostrophic winds accelerate with height due to the equatorward temperature increase, reaching core speeds of 25-100 m/s in the polar jet at approximately 9-11 km altitude near the tropopause. This height-dependent speedup continues until the tropopause level, where the temperature gradient diminishes, forming a jet core that is roughly 5-10 km thick vertically and 1,000-2,000 km wide horizontally. As per the thermal wind equation, this shear reflects the integrated effect of horizontal temperature contrasts, concentrating the strongest winds aloft.^[22]^[10] Seasonal variations in jet stream intensity stem from changes in meridional temperature contrasts, with stronger gradients in winter enhancing thermal wind shear and thus jet speeds, while summer warming reduces these contrasts and weakens the jets. In winter, polar jets achieve speeds of 25-100 m/s, positioned between 30-60°N/S, whereas in summer, speeds drop to 0-45 m/s, and the jets shift poleward to around 40-45° latitude, often merging into a single band. These patterns are evident in long-term reanalysis data, underscoring the thermal wind's role in modulating jet persistence and location.^[22] Under climate change, increasing global temperatures are projected to enhance meridional temperature gradients in the upper troposphere, leading to stronger thermal wind shear and faster upper-level jet streams, particularly in the extratropics, with increases of about 2.1% per Kelvin of warming as of studies through 2023.^[25] Thermal wind also influences Rossby wave propagation within jet streams, where the vertical shear modulates wave amplitudes by altering the mean flow and baroclinicity that support planetary-scale meanders. These meanders, or Rossby waves, form troughs and ridges in the jet, with thermal wind-driven wind variations affecting wave growth and eastward propagation speeds, typically guiding mid-latitude weather systems.^[10]^[26]

Frontogenesis Processes

Frontogenesis involves the intensification of horizontal temperature gradients, which, through the thermal wind relation, directly enhances the vertical shear of the geostrophic wind and drives ageostrophic circulations that further sharpen frontal boundaries. Horizontal contraction of these gradients, often via advective processes, increases the magnitude of the temperature gradient vector (∇T), thereby amplifying the thermal wind shear and promoting baroclinic instability in the atmosphere. This mechanism results in a feedback loop where the strengthened shear induces convergent motions, concentrating isentropes and accelerating front development.^[27] Kinematic frontogenesis specifically arises from deformation fields in the horizontal wind, characterized by stretching in one direction and contraction in the perpendicular direction, which deform and amplify ∇T. As the temperature contrast sharpens, the thermal wind equation dictates a corresponding increase in the vertical derivative of the horizontal wind (∂V/∂z), boosting the baroclinicity and vorticity along the front. This process is particularly effective in regions of geostrophic deformation, where the non-divergent flow initially tightens gradients, setting the stage for subsequent ageostrophic enhancements.^[28] A prominent example occurs during the sharpening of cold fronts in extratropical cyclones, where the thermal wind aligns parallel to the frontal zone, generating substantial vertical shear that aligns with the temperature contrast. In such systems, this shear contributes to intense precipitation and dynamic lifting along the front. The alignment enhances the front's slope and narrows its width, exemplifying how thermal wind imbalances sustain the cyclone's vigor.^[29] The Sawyer-Eliassen equation conceptually links these dynamics by modeling the cross-frontal ageostrophic circulation as a response to geostrophic forcing, including thermal wind imbalances that drive thermally direct circulations. This circulation converges air toward the frontal surface on the warm side and diverges on the cold side, further intensifying ∇T and the associated shear without requiring explicit derivation of the thermal wind itself. Such circulations are essential for maintaining frontogenetic tendencies in balanced flows.^[27]

Thermal Advection Effects

Horizontal temperature advection alters thermal wind profiles by changing the horizontal temperature gradient over time, which directly influences the direction and magnitude of the vertical wind shear. In regions of warm advection, such as southwesterly geostrophic flow transporting warmer air equatorward of a front, the thermal wind vector veers clockwise with height, reflecting a rotation in the shear direction due to the modified gradient. Conversely, cold advection, often associated with northeasterly flow, causes the thermal wind to back anticlockwise with height. This directional change arises because advection perturbs the geostrophic balance, prompting ageostrophic circulations to adjust the temperature field and restore equilibrium.^[30]^[31] The process involves advective modifications to the temperature gradient that rotate the thermal wind vector, typically resulting in 45-90° turns over the depth of the troposphere during active synoptic conditions. These turns are conceptually linked to Q-vector convergence, where the divergence pattern of Q-vectors—representing the advection of thermal wind by geostrophic flow—drives vertical motions that further influence the shear profile. In warm advection scenarios, Q-vector convergence in the lower troposphere enhances ascent, which can amplify the veering effect by reinforcing the temperature perturbations.^[32]^[33] A prominent synoptic example occurs in mid-latitude cyclones, where the warm conveyor belt advects moist, warm air northward ahead of the warm front, strengthening the thermal wind while turning it clockwise with height to maintain baroclinicity. This advection-driven veering contributes to the cyclone's intensification and guides its track by aligning the upper-level jet with the surface low. Such dynamics highlight how thermal advection not only alters local wind profiles but also shapes larger-scale weather evolution, occasionally initiating frontogenesis through gradient sharpening.^[34]^[35]

Observational and Diagnostic Uses

Measurement Techniques

Radiosonde observations offer a direct method for measuring vertical profiles of wind and temperature as functions of height or pressure, enabling the computation of thermal wind as the vertical shear in geostrophic wind between standard pressure levels, such as 850 hPa and 200 hPa. These balloon-borne instruments ascend through the atmosphere at approximately 5-6 m/s, collecting data at resolutions of 6-10 seconds for temperature and pressure, and deriving winds from GPS positioning or tracking. The resulting profiles allow application of the thermal wind relation to quantify horizontal temperature gradients driving upper-level wind changes, with validations showing good agreement between observed and geostrophically computed shears in mid-latitude soundings.^[36]^[37]^[38] Satellite-based techniques estimate thermal wind by retrieving atmospheric temperature profiles from infrared sounders, such as the TIROS Operational Vertical Sounder (TOVS), which measure brightness temperatures to infer horizontal temperature gradients via hydrostatic and thermal wind balance. These retrievals incorporate variational methods to enforce mass conservation, combining temperature data with planetary boundary layer parameterizations for wind estimation across the troposphere. Validations against ground stations indicate mean absolute errors of approximately 1 m/s in polar regions, demonstrating utility for large-scale thermal wind mapping where direct observations are sparse.^[39]^[40] Reanalysis datasets like ERA5 from the European Centre for Medium-Range Weather Forecasts integrate global observations and model outputs to provide consistent fields of geopotential height, temperature, and winds on pressure levels, facilitating thermal wind computation through differences in geostrophic winds between levels such as 1000 hPa and 500 hPa. Tools like Metview process ERA5 geopotential data to derive thickness fields (e.g., 500-1000 hPa), from which thermal wind vectors are calculated assuming geostrophic balance, enabling vertical integration for synoptic-scale maps. This approach supports global monitoring of thermal wind patterns, with ERA5's 31 km horizontal resolution and hourly temporal coverage enhancing diagnostic applications over decades.^[41]^[42] Modern advancements include Doppler lidars for high-resolution wind and temperature profiling in the middle atmosphere, as demonstrated in post-2010 field campaigns like the Aeolus Calibration/Validation efforts, where airborne systems measured profiles up to 80 km with wind accuracies of 0.6 m/s at mid-altitudes. These instruments use Rayleigh scattering at 532 nm to derive line-of-sight winds and hydrostatic temperatures, allowing thermal wind estimation in targeted observing scenarios such as polar vortex studies. Complementarily, GPS dropsondes deployed from research aircraft in campaigns like NASA's Genesis and Rapid Intensification Processes (GRIP) provide high-vertical-resolution (5 m) thermodynamic and wind profiles from flight level to the surface, particularly in tropical cyclones, supporting thermal wind computations for boundary layer shear analysis.^[43]^[44]^[45]

Diagnostic Implications in Weather Systems

Thermal wind serves as a key diagnostic tool in forecasting atmospheric circulations by quantifying baroclinicity, where strong vertical shear in the geostrophic wind indicates regions of significant horizontal temperature gradients conducive to cyclone development. In operational meteorology, this is often assessed through 500–1000 hPa thickness maps, which reflect the mean temperature in that layer and relate directly to the thermal wind magnitude via the hypsometric equation; tighter spacing of thickness contours signals enhanced baroclinicity and potential for vorticity-driven cyclone intensification.^[46] For instance, decreasing thickness gradients in such maps highlight areas where thermal wind strength promotes rapid extratropical cyclone growth, aiding forecasters in predicting storm tracks and intensity changes.^[46] Deviations from expected thermal wind balance reveal ageostrophic components in weather systems, signaling processes such as frontolysis or storm intensification. In frontal zones, thermal wind imbalances ahead of cold fronts arise from asymmetric advection of momentum and potential temperature, leading to mesoscale ageostrophy that can weaken frontal structures during frontolytic phases or enhance low-level jets contributing to precipitation.^[47] Similarly, in extratropical cyclones, observed thermal winds significantly stronger than model forecasts (e.g., ratios exceeding 1.3) indicate underpredicted synoptic-scale thermal gradients, often resulting in errors in storm intensity and steering flows, as ageostrophic accelerations disrupt geostrophic balance.^[48] In a climate context, long-term shifts in thermal wind due to Arctic amplification have weakened mid-latitude jet streams, with post-2000 studies linking reduced meridional temperature gradients to decreased vertical wind shear and storm track intensity. This weakening, evident in a 15% decline in synoptic storm kinetic energy since 1979, fosters persistent weather patterns that exacerbate mid-latitude extremes like cold outbreaks and heatwaves.^[49] Such changes, driven by faster Arctic warming (2–4 times the global rate), diminish the poleward temperature gradient, thereby reducing thermal wind and altering circulation dynamics over decades.^[50] A notable case study is the 1987 Great Storm over Europe, where intense thermal wind shear from strong baroclinicity—stemming from a sharp temperature contrast between cold polar air and warmer Atlantic air—drove rapid cyclone deepening and unexpected track deviations, leading to forecast errors in operational models. This event underscored thermal wind diagnostics in identifying explosive cyclogenesis, as the enhanced shear amplified upper-level divergence and surface winds exceeding 100 km/h, causing widespread damage.^[51]

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Jun 13, 2018 · Although the principal thermal wind used in operational meteorology is that through the layer between 1000 hPa and 500 hPa, in fact thermal wind ...
[46]
https://resources.eumetrain.org/ePort_MapViewer/docs/thickness_and_its_uses.pdf
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[PDF] using the thermal wind relationship to improve offshore and coastal ...
A thermal-wind observation that is significantly stronger than the numerical model forecast value can indicate significant errors in the modeled cyclone ...
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The influence of Arctic amplification on mid-latitude summer circulation
Aug 20, 2018 · This weakening has been detected in the westerly jet (following the thermal-wind balance), the total kinetic energy of synoptic storm ...
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Arctic Amplification Weakens the Variability of Daily Temperatures ...
Arctic amplification (AA) reduces meridional temperature gradients (dT/dy) over the northern mid-high latitudes, which may weaken westerly winds.
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BBC NEWS | Science/Nature | Lessons learned from Great Storm
Oct 14, 2007 · "In 1987, there was obviously a very strong temperature gradient." These gradients are caused by the difference in temperature in equatorial ...