ATMS 310 Thermal Wind

The thermal wind is defined as the vector difference between the geostrophic winds at two levels. It is not really a wind at all, just a measure of the shear of the geostrophic wind. But there are good reasons for considering the geostrophic wind; mainly, it provides a convenient way of connecting the structure of the temperature field to the wind field.

To see the connection, first recall the hypsometric equation. This says that the thickness between any two pressure levels is proportional to the mean temperature within that layer: RT z = δδ ln p (1) g

If the geostrophic wind is increasing with height, then the horizontal pressure gradient must also be increasing with height. If the pressure gradient is increasing in the positive x-direction, the temperature gradient must also be increasing in that direction.

In the figure above, the thickness of the atmosphere at x2 is greater than the thickness at x1. We know through the hypsometric equation that the mean temperature between pressure levels po and po+2dp must be greater in the x2 column than the x1 column. This causes the horizontal pressure gradient to change more rapidly over x2 as height increases, resulting in a stronger geostrophic wind (dark arrows).

The thermal wind equation describes how much the geostrophic wind will change with height:

∂Vg R v ∇×−= Tk (2) ∂ ln p f p (a) (b)

where (a) is the local rate of change of the geostrophic wind with pressure, which depends on (b), the horizontal temperature gradient. If the temperature is constant across the pressure level, then the geostrophic wind is independent of height and motion in the atmosphere is constrained.

The thermal wind vector points parallel to the isotherms with warm air to the right facing downstream:

Direction changes of the geostrophic wind with height can be used to find if warm air advection (WAA) or cold air advection (CAA) is occurring in the layer. If the wind is veering (turning clockwise) with height, we have a WAA situation. If the wind is backing (turning counter-clockwise) with height, CAA is occurring.

Barotropic vs. Baroclinic Atmosphere

A barotropic atmosphere is one in which density is only a function of pressure:

ρ = ρ p)(

A barotropic atmosphere puts a severe constraint on atmospheric motion. Constant pressure and density surfaces are parallel. Also, constant pressure surfaces are isothermal for an Ideal Gas (which the atmosphere is usually considered). This makes the geostrophic wind independent of height, as shown in the thermal wind equation (thus there would not be a thermal wind, since Vg does not change with height).

The deep tropics most closely approximates a barotropic atmosphere.

A baroclinic atmosphere is one in which density is a function of pressure and temperature:

ρ = ρ Tp ),(

Now that the temperature field can be independent of the pressure field, the thermal wind equation applies and the geostrophic wind can change with height. Any models that deal with any locations besides the deep tropics must assume baroclinicity to yield reasonable results.

Vertical Motion – Kinematic and Adiabatic Methods

Vertical motions in the atmosphere are usually subtle, on the order of 1 cm/s. This is well beyond our current accuracy in measuring wind speeds, which is on the order of 1 m/s. But there are a couple of techniques that can be used to deduce the vertical velocity from the large-scale fields that we can measure.

Kinematic Method This method is based on the continuity equation. If the isobaric form of the continuity equation is integrated,

p ⎛ ∂u ∂v ⎞ ωω pp )()( −= ⎜ + ⎟ dp (3) s ∫⎜ ∂x ∂y ⎟ ps ⎝ ⎠ p (a) (b) (c) where a) is the vertical velocity at the pressure level p (remember that negative values of ω indicate upward motion), b) is the vertical velocity at a reference pressure level, and c) horizontal divergence of the wind between ps and p. Equation (3) may also be written as:

⎛ ∂ u ∂ v ⎞ ωω −+= pppp )()()( ⎜ + ⎟ (4) s s ⎜ ∂x ∂y ⎟ ⎝ ⎠ p where the angled brackets represent a pressure weighted average for the layer (see Holton). Equation (4) can be written in terms of w, the vertical velocity in (x,y,z,t) coordinates: ρ zwz )()( − pp ⎛ ∂ u ∂ v ⎞ sss ⎜ ⎟ zw )( = − ⎜ + ⎟ z ρρ )()( gz ⎝ ∂x ∂y ⎠ The advantage of the kinematic method is that vertical velocity can be derived from just knowledge of the large-scale horizontal winds. However, just a 10% error in one of the wind components can produce a 100% error in the wind divergence calculation. This is a big concern, and is the main reason why the kinematic method is not used operationally.

Adiabatic Method The adiabatic method is based on the thermodynamic equation. It’s main draw is that it only requires height and temperature data to calculate the vertical velocity. However, if there is strong diabatic heating in the area, the adiabatic method quickly falls apart. In addition, it is sometimes difficult to determine the local rate change of temperature term. The equation for the adiabatic method is:

−1 ⎛ ∂T ∂T ∂T ⎞ ω = S p ⎜ + u + v ⎟ (5) ⎝ ∂t ∂x ∂y ⎠ where Sp is the static stability parameter, which can be calculated as:

RT ∂T S p −≡ ()d Γ−Γ≡ / ρg (6) p pc ∂p where Γd is the dry adiabatic lapse rate (9.8°C/km) and Γ the actual lapse rate at that level.