Applications of the Basic Equations Chapter 3

Home , Temperature gradient, Thermal wind

Part 3: The Thermal Wind Question: Is there a relationship between wind speed and temperature?

It turns out that there is a close link between vertical wind shear (vertical gradients of horizontal wind speed) and layer thickness, which is governed by temperature.

Paul Ullrich Applications of the Basic Equations March 2014 300 hPa Wind Speed

Question: Where are the strongest temperature gradients?

Paul Ullrich Applications of the Basic Equations March 2014 850 hPa Temperature

Wind speeds appear to be largest where temperature gradients in lower layers are strongest.

Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind

Defnition: The thermal wind is a vector diﬀerence between the geostrophic wind at an upper level and a lower level.

Figure: Thickness of layers related to temperature, causes a tilt in the pressure surfaces.

Change in magnitude of horizontal gradient of pressure then leads to vertical wind shear. Cold Warm

Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind

Emphasis: The thermal wind is not a real wind, but a vector diﬀerence.

The thermal wind vector points such that cold air is to the left and warm air is to the right, parallel to isotherms (in the northern hemisphere). Cold air is to the right and warm air is to the left in the southern hemisphere.

Cold Warm

Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic Wind In Pressure Coordinates

1 @p Recall: The geostrophic wind is the component u = of the real wind which is governed by geostrophic g f⇢ @y balance. On constant height surfaces, it is defned 1 @p to satisfy v =+ g f⇢ @x

In pressure coordinates, this relationship takes on the analogous form:

1 @ u = g f @y 1 @ v =+ g f @x

Paul Ullrich Applications of the Basic Equations March 2014 1 @ 1 @ Geostrophic u = v = g f @y g f @x Wind ✓ ◆p ✓ ◆p

Diﬀerentiate with respect to p

@u 1 @ @ @v 1 @ @ g = g = @p f @y @p @p f @x @p

@ @z 1 RT Hydrostatic = g = = On constant p Relationship @p @p ⇢ p surfaces

@ug R @T @v R @T Thermal wind = g = relationship: @p pf @y @p pf @x ✓ ◆p ✓ ◆p

Paul Ullrich Applications of the Basic Equations March 2014 @ug R @T @v R @T Thermal wind = g = relationship: @p pf @y @p pf @x ✓ ◆p ✓ ◆p

This relationship links the horizontal temperature gradient with the vertical wind gradient (vertical shear).

Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind

@u R @T Thermal wind g = relationship: @p pf @y ✓ ◆p @v R @T g = The thermal wind itself @p pf @x ✓ ◆p is a vector diﬀerence.

@ug R @T @vg R @T Rewrite = = @(log p) f @y @(log p) f @x ✓ ◆p ✓ ◆p R @ T p Integrate u = u (p ) u (p )= h i log 1 T g 2 g 1 f @y p ✓ ◆p ✓ 2 ◆ R @ T p Thermal Wind v = v (p ) v (p )= h i log 1 T g 2 g 1 f @x p ✓ ◆p ✓ 2 ◆

Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind R @ T p u = u (p ) u (p )= h i log 1 T g 2 g 1 f @y p ✓ ◆p ✓ 2 ◆ R @ T p v = v (p ) v (p )= h i log 1 T g 2 g 1 f @x p ✓ ◆p ✓ 2 ◆

Example: Thermal wind vT between 500 hPa and 1000 hPa

ug (p2 = 500 hPa)

uT thermal wind

ug (p1 = 1000 hPa)

Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind

Alternate form of thermal wind, written in terms of geopotential height (obtained from hypsometric equation):

R p 1 u = d k T log 1 = k ( ) T f ⇥rph i p f ⇥rp 2 1 ✓ 2 ◆ 1 @ 1 @ u = ( ) v = ( ) T f @y 2 1 T f @y 2 1

Index “1” indicates lower level, “2” indicates upper level.

Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind R @ T p u u (p ) u (p )= h i log 1 T ⇡ g 2 g 1 f @y p ✓ ◆p ✓ 2 ◆ R @ T p v v (p ) v (p )= h i log 1 T ⇡ g 2 g 1 f @x p ✓ ◆p ✓ 2 ◆

Note that thermal wind always points parallel to lines of constant layer temperature:

R p @ T @ T @ T @ T u T = log 1 h i h i + h i h i =0 T · rh i f p @y @x @x @y ✓ 2 ◆

Paul Ullrich Applications of the Basic Equations March 2014 Thermal Wind Thermal wind always points parallel to lines of constant temperature (and lines of constant layer thickness).

Defnition: Veering winds are Cold defned by clockwise rotation Thermal wind of the geostrophic wind with Lower level height and are associated with geo. wind Upper level warm air advection. geo. wind Warm

Defnition: Backing winds are Cold Upper level defned by counter-clockwise geo. wind Lower level rotation of the geostrophic wind geo. wind with height and are associated with cold air advection. Thermal wind Warm

Paul Ullrich Applications of the Basic Equations March 2014 The thermal wind determines the relationship between meridional temperature gradients and zonal winds.

Question: Given zonal mean temperature below, where are zonal jets?

@T Wintertime DJF Summertime @y ?

y Paul Ullrich Applications of the Basic Equations March 2014 The thermal wind determines the relationship between meridional temperature gradients and zonal winds.

Question: Given zonal mean temperature below, where are zonal jets?

@T Wintertime DJF Summertime @y ? - - - - +

- + y Paul Ullrich Applications of the Basic Equations March 2014 Question: Given zonal mean temperature below, where are zonal jets? @u R @T g = @p pf @y ✓ ◆p @T Wintertime DJF Summertime @y ? - - - + @u g ? @p - - - + + +

- - + - y Paul Ullrich Applications of the Basic Equations March 2014 Question: Given zonal mean temperature below, where are zonal jets? @u R @T g = @ug @ug @p pf @y ✓ ◆p @z ⇠@p @T Wintertime DJF Summertime @y ? - + - - @u g ? @z - + - - + -

- + + + y Paul Ullrich Applications of the Basic Equations March 2014 Question: Given zonal mean temperature below, where are zonal jets? @u R @T g = @ug @ug @p pf @y ✓ ◆p @z ⇠@p Wintertime DJF Summertime

y Paul Ullrich Applications of the Basic Equations March 2014