Feeds:
Posts

## Thermal Wind

One of the fundamental characteristics of a planetary atmosphere is its wind distribution with height. Shown in the opening figure is the east-west component (or the u component) of wind speed on a latitude-height grid. The vertical axis is pressure, which decreases upward with height, since there is less air above you as one progresses higher in the air column.

One of the most noticeable features of the figure is that winds become more westerly (i.e., stronger towards the east) with height. The question motivating this post is what drives this observed phenomenon, and can we come up with a relationship between vertical wind shear (or the change in wind speed/direction with height) and horizontal temperature gradients? It is certainly not intuitive that such a relationship would exist. By the end of this post, we will conclude that the presence of westerly vertical shear is a direct consequence of the uneven solar heating of the Earth (more heating at the equator) coupled with the dominance of something known as the thermal wind balance, particularly in mid-latitudes. Broader application of this thermal wind balance will arise in future posts.

I must back up for a moment and lay out a few fundamental physical relationships that will set the stage for our exploration of thermal wind. The physical steps I will outline here may seem random, but hopefully it will all come together in the end.

Consider first the question of how “thick” a layer of atmosphere is between two pressure surfaces. That is, pick your two favorite two pressure levels above the surface (say 800 millibars as the level closer to the surface, and 400 millibars further up in the atmosphere). Then, what is the physical distance between those two locations in the atmosphere? A starting point to explore this question is the hydrostatic equation, which can be derived from a simple force balance: Gravity is always trying to pull air molecules close to the surface. In contrast, there exists a vertical pressure gradient which compels air molecules at higher pressure (close to the surface) upward to regions of lower pressure (further above the surface). The competing influence of these two effects is why our atmosphere does not fly away into space or get dragged down to a thin layer near the surface. Hydrostatic equilibrium is obeyed to very good accuracy in planetary atmospheres, and even in the sun for example. The mathematical expression for this balance is:

$\displaystyle \frac{\partial p}{\partial z} = - \rho g$

the left hand side of the equation is the differential of pressure with respect to height, $\rho$ is density, and $g$ is the acceleration due to gravity. We also know of another, more familiar fundamental law that emerges from thermodynamics: the equation of state for an ideal gas, which is $p = \rho RT$, where R is a gas constant appropriate to the atmosphere in consideration and T is temperature. We can combine these two equations to yield,

$\displaystyle \frac{\partial p}{\partial z} = - \frac{pg}{RT}$

We can integrate this relationship between two pressure levels, which gives a relation:

$\displaystyle \Delta z = z_{2} - z_{1}= \frac{RT}{g} ln(\frac{p_{1}}{p_{2}})$

where p1 > p2 and the height z2 > z1. Since R and g are constants, and we have fixed what pressure levels we are interested in, we end up with the conclusion that the thickness between the two pressure levels is directly proportional to the temperature of the layer.

As a next step in this analysis, we have learned in high school that horizontal (east- west, or north-south) wind is caused by pressure gradients. Wind is directed toward regions of lower pressure, but is deflected by the coriolis effect that occurs in a rotating frame of reference. In particular, we call the north-south direction the y direction and the east-west direction the x direction; the velocity in those two directions (the wind) is the v and u component, respectively. Ignoring a few small terms, the accelerations are given by:

$\displaystyle \frac{du}{dt} = -\frac{1}{\rho} \frac{dp}{dx} + fv$
and,
$\displaystyle \frac{dv}{dt} = -\frac{1}{\rho} \frac{dp}{dy} - fu$

The term f is called the Coriolis parameter, which is proportional to the sine of the latitude, and also increases for higher planetary rotation rates.

In the mid and high latitudes, the influence of the coriolis effect and the pressure gradients are typically an order of magnitude larger than horizontal accelerations. In particular, when one moves away from the surface (where friction is important, one of those “small terms” mentioned before), then we can define a force balance between the horizontal pressure gradient and Coriolis effect. This gives rise to the geostrophic component of the total wind, which in many regions of the planet constitutes a substantial fraction (or nearly all) of the total wind field. The deviation from geostrophy is usually very small about ~1 km. It is also convenient to eliminate density from these equations and define a term called geopotential, $d \phi = g dz$. The total geostrophic wind component can be expressed as blowing parallel to lines of constant geopotential heights (or heights), $\overrightarrow{V_{g}} = (\widehat{k}/f) \; x \; \nabla \phi$ (the vector $\overrightarrow{V_{g}}$ is the total geostrophic wind and $\widehat{k}$ is the vertical direction that one applies the cross-product to, which defines the wind direction. For those uncomfortable with such vector notation, the graphical expression of the above equation is shown below (see description in caption):

Vertical cross-section of a horizontal contrast in temperature (or equivalently, geopotential height). Temperature increases to the east. Lines of constant pressure are shown for 1000, 800, and 600 mb. Because of the thickness equation, these lines slope downward to the region of cold temperature. The circles indicate the wind field. Note that the wind direction is into of the screen away from the viewer, and the size of the circle represents the magnitude of the wind speed (increasing with height).

The vertical gradient of the geostrophic wind is:

$\displaystyle \frac{\partial \overrightarrow{V_{g}}}{\partial p} = \frac{\widehat{k}}{f} \; x \; (\nabla \frac{\partial \phi}{\partial p})$

The Conclusion

Finally, because the layer-averaged temperature is directly related to the layer-averaged thickness between layers, we can write an expression for the vertical shear in the geostrophic wind. For both the east-west geostrophic wind ($u_{g}$) variation with pressure, and the north-south geostrophic wind component ($v_{g}$), we write:

$\displaystyle \frac{\partial u_{g}}{\partial p} = \frac{R}{fp} \frac{\partial T}{\partial y}$

and,

$\displaystyle \frac{\partial v_{g}}{\partial p} = - \frac{R}{fp} \frac{\partial T}{\partial x}$

This confirms the rather remarkable conclusion that the vertical gradient of the horizontal winds is proportional to the horizontal temperature gradient! The physics is straightforward from the second figure: Increasing slopes of constant pressure surfaces, which is related to gradients in geopotential height, which in turn is determined by gradients in temperature. Because the geostrophic wind increases in magnitude as one tightens pressure gradients, the increased slopes in Figure 2 imply increasing geostrophic wind. See Holton’s “Introduction to Dynamic Meteorology” or Jonathan Martin’s “Mid-Latitude Atmospheric Dynamics” for similar presentations and a more complete discussion/application of thermal wind balance.

From the first equation for example, one “broad” application of the first of these two equations is that temperature decreases poleward. That is dT/dy is negative. Therefore dug/dp is negative (alternatively, dug/dz is positive if you intuit things better in height coordinates). This confirms our model and observational result that winds become more westerly in the mid latitudes. In the region between the equator and the Sahel (near 30 N) in Africa, temperatures actually increase poleward and an easterly jet is found.

Note that in the oceans, an equation of state can be used to give density gradients in terms of temperature/salinity variations. Neglecting salinity gradients, if ocean temperatures decrease toward the poles, currents will become increasingly westward as a function of depth.

Application of the thermal wind has extremely important use for meteorologists and atmospheric dynamicists. It has been used extensively by atmospheric scientists (and at least one study used its application in order to use wind measurements as a proxy for temperature trends in the upper, tropical atmosphere, given the large uncertainties that have plagued the robust detection of this phenomenon- the relative warming of the upper atmosphere relative to the surface in a warming world has strong implications for tropical dynamics and stability). I will return to application of the thermal wind equation in later posts on the general circulation of our atmosphere.

A couple of notes:

The thermal wind equation can be applied to other atmospheres as well, although this post is a bit Earth-centric in that geostrophic balance is not always a good approximation. In particular, the Coriolis component of that balance becomes much less important at lower rotation rates (as on Venus, or tidally-locked exoplanets) and other terms become more important. Geostrophy relates the 3-dimensinal structure of the winds and temperatures at a given time but says nothing about the flow’s time evolution (note in this post that we removed the acceleration term from the momentum equations in order to come up the geostrophic balance, which lost any time dependent terms). In a rapidly rotating atmosphere, both the temperatures and winds will generally evolve together, and maintain approximate geostrophic balance as they do so. On Venus, in contrast, the curvature term $u^{2}/r \; tan( \theta )$ (latitude $\theta$ and planet radius $r$) is the dominant force term along with the pressure gradient force. This yields a “cyclostrophic balance” that neglects the coriolis term (this is also important in tornadoes, for example). One can come up with a thermal wind version that is applicable to a flow predominately in cyclostrophic balance, and in fact the temperature gradient becomes directed toward the equator regardless of whether the wind shear is westerly or easterly with height, in this regime. It is also worth adding that it is often preferable to use the so-called virtual temperature in determining layer thickness as opposed to real temperature. This is a minor correction for the presence of water vapor in the atmosphere and its impact on air density (this way one could retain the dry gas constant R in the thermodynamic relation). It’s a very small effect however.