Chapter 14: Polar Weather

What are some unique aspects of the polar atmosphere relative to the mid- latitudes?

Katabatic Winds

Katabatic wind – a wind that flows downslope under the influence of gravity

Katabatic winds are also known as drainage winds and slope flows.

Example: Continent-wide Antarctic katabatic winds

Example: Extreme Antarctic katabatic winds – the Adelie Coast What conditions are necessary for katabatic winds to occur?

- sloped surface

- cooling of air in contact with the surface

How is air in contact with the Antarctic ice sheet cooled?

What type of atmospheric temperature profile results from this cooling?

Fall line – local downslope direction

What is the direction of Antarctic katabatic winds relative to the fall line?

Can katabatic winds develop in other locations?

If so, how do they differ from Antarctic katabatic winds?

Some mean wind definitions

For a time series consisting of N wind observations we can define:

v 1 N Scalar mean wind speed: V = WS N е i i=1 where WSi is the wind speed of the ith wind observation.

The horizontal wind components (u and v) can be calculated from the wind speed (WS) and wind direction (WD) as:

u = - WS ҙ sin(WD) v = - WS ҙ cos(WD) where u is the zonal (east/west) wind component and v is the meridional (north/south) wind component.

The mean zonal and meridional winds can be calculated as:

1 N 1 N u = u v = v е i е i N i=1 N i=1

Resultant wind speed - Vector averaged wind speed

Resultant WS = u 2 + v 2

The resultant wind direction (vector averaged wind direction) is given by:

180 - 1жv ц WD = 90 - tan з ч + WD0 p иu ш

where WD0 = 180° for u >0 and WD0 = 0° for u <0

This equation is not valid for u =0. In this case WD = 180° for v >0 (south wind) and WD = 360° for v <0 (north wind).

If u =0 and v =0 the wind is calm and the wind direction is defined as 0°.

Example: Scalar and resultant wind speeds – a simple case

Calculate the scalar and resultant wind speed for:

-1 WS1 = 10 m s , WD1 = 360° -1 WS1 = 10 m s , WD1 = 180°

Directional constancy (DC) – the ratio of the vector mean wind speed to the scalar mean wind speed u 2 + v 2 DC = 1 N v V е i N i=1

The directional constancy is a measure of how much the wind direction varies over time at a particular location.

What is the directional constancy of the wind from the previous example? Example: Directional constancy of Antarctic katabatic winds

Data: Wind observations from Terra Nova Bay for March 1997 (available on CD-ROM)

Key features of Antarctic katabatic winds:

- large wind speeds

- large directional constancy

- wind direction 10 to 50° to the left of the local fall line

What are the dynamics of Antarctic katabatic winds?

Consider mid-winter conditions in the interior of the Antarctic ice sheet:

In order to study the dynamics of katabatic winds we will assume the atmosphere over the Antarctic ice sheet consists of two layers, each of constant density () and potential temperature ().

We will assume that the lower layer is of constant depth, and that the interface between the lower and upper layers is parallel to the underlying ice slope.

Since the air near the surface is cooling we will take bot < top and bot > top Start with the horizontal components of Navier-Stokes equations and assume:

- the flow is steady state (¶ ¶t = 0)

- the advection terms can be neglected

- the viscous force is important and can be approximated as CDVu in the x- direction and CDVv in the y-direction

- The x-axis points in the downslope direction and the y-axis points to the left of the fall line.

With these assumptions the horizontal equations of motion are:

1 ¶p 0 = - + fv - C Vu r ¶x D 1 ¶p 0 = - - fu - C Vv r ¶y D

What does each term in these equations represent?

Use the hydrostatic equation to calculate the pressure at point A in the figure on the previous page.

¶p = - r g ¶z

pB zB ¶p = - r g¶z Ю т т bot pA zA

pB - pA = - r bot g(zB - zA )

pA = pB + r bot g(zB - zA )

The pressure at point D is given by:

pD = pC + r top g(zC - zD )

Use these expressions for the pressure at points A and D to calculate the downslope (x-direction) pressure gradient force.

1 ¶p 1 p - p - » - D A r ¶x r Dx 1 (p + r gh) - (p + r gh) » - C top B bot r Dx

1 жp - p (r top - r bot )ghц » - з C B + ч r з Dx Dx ч и ш

where r is the mean density of the two layers and Dx is the horizontal distance between points A and D

Define the horizontal pressure gradient force between points B and C as

1 p - p 1 ¶p - C B » - r Dx r ¶x top

This term is often referred to as the ambient or background pressure gradient force.

From the geometry illustrated in the figure h Dx = ¶z ¶x is the terrain slope.

The x-component of the pressure gradient force can then be written as:

1 ¶p 1 ¶p (r top - r bot ) ¶z - = - - g r ¶x r ¶x top r ¶x

rtop - r bot qtop - qbot Dq This can be rewritten by noting that: » = , where q is r q q

the mean potential temperature of the two layers and Dq is the strength of the temperature inversion.

1 ¶p 1 ¶p Dq ¶z - = - - g r ¶x r ¶x top q ¶x

ж Dq ¶z ц Katabatic force з- g ч – the component of the horizontal pressure и q ¶xш gradient force that arises solely due to the presence of a potential temperature inversion over sloping terrain

What is the sign of this term for a potential temperature inversion over terrain that slopes down in the positive x direction?

What will cause this term to be zero?

Since our coordinate system is oriented such that there is no terrain slope in the y-direction the horizontal pressure gradient force in the y-direction is given by:

1 ¶p 1 ¶p - = - r ¶y r ¶y top

Using these expressions for the horizontal components of the pressure gradient force the horizontal equations of motion for katabatic winds are:

1 ¶p Dq ¶z Downslope momentum equation: 0 = - - g + fv - CDVu r ¶x top q ¶x

1 ¶p Cross-slope momentum equation: 0 = - - fu - CDVv r ¶y top

Often the background pressure gradient force terms are neglected and the katabatic wind results from a balance between the katabatic force, the Coriolis force, and the frictional force. What wind direction results if we neglect the frictional force (Fr), the Coriolis force (CoF), or consider all three forces (katabatic, Coriolis, and frictional) balancing?

We can solve our horizontal momentum equations that describe katabatic winds for the katabatic wind speed (V) and deviation angle ().

Deviation angle (): The angle of the wind from the terrain fall line, with positive values taken for winds that are directed to the left of the fall line

0.5 2 2 f ж f жg Dq ¶z ц ц + з + 4 ч з 2 з ч ч CD C D и f q ¶xш cosy = и ш жg Dq ¶z ц 2з ч и f q ¶xш

g Dq ¶z V = siny f q ¶x 0.5 ж g Dq ¶z ц = з- cosy ч и CD q ¶x ш u = V cosy v = V siny

Example: Calculate the downslope and cross-slope components of an idealized katabatic flow

Given:

 = 75°S bot = 230 K top = 250 K ¶z ¶x = -0.002 -5 -1 CD = 1.67x10 m

Assume that the background pressure gradient force is negligible.

What is the direction of the katabatic wind relative to the fall line?

What is the katabatic wind speed?

What are the u and v components of this wind?

Antarctic katabatic winds and the circulation of the high latitudes of the Southern hemisphere Barrier Winds

What happens when stably stratified air is directed towards a topographic barrier?

V Fr = 0.5 Froude number: ж Dq ц зgH ч и q ш

V – wind speed towards barrier H – height of barrier  - potential temperature difference between top and bottom of barrier  - potential temperature of flow approaching the barrier

The Froude number is proportional to the ratio of the kinetic energy of the flow to the potential energy required for the flow to pass over the barrier.

For Fr>1 the flow will pass over the barrier

For Fr<1 the flow is blocked by the barrier

What conditions lead to Fr>1 and Fr<1?

Example: Flow across the Ross Ice Shelf, Antarctica

Given:

- Easterly wind of 10 m s-1 directed towards the Transantarctic Mountains

- Elevation of Transantarctic Mountains = 2000 m

- Potential temperature of near surface air over the Ross ice shelf = 260 K

- Potential temperature difference between the top and bottom of the Transantarctic Mountains = 10 K

What is the Froude number for these conditions? What does this imply about the ability of this flow to pass over the Transantarctic mountains?

What is the atmospheric response to a blocked flow (Fr<1)?

Consider an atmosphere with two layers, each with uniform density and potential temperature

For these layers we will take bot < top and bot > top (stable stratification) h is the undisturbed depth of the lower layer and h is change in depth of the lower layer due to flow blocking by the terrain.

Using the hydrostatic equation the pressure at points A and D can be calculated as:

pA = pB + r bot (h + Dh)g pD = pC + r bothg + r topDhg

The pressure difference between points A and D is given by:

pA - pD = pB + r bot (h + Dh)g - pC - r bothg - r topDhg

= pB - pC + r bothg - r bothg + r botDhg - r topDhg

= pB - pC + (r bot - r top )Dhg

If there is no horizontal pressure difference between points B and C the pressure difference between points A and D reduces to:

pA - pD = (r bot - r top )Dhg

Is this pressure difference positive or negative?

Example: Calculate the pressure difference for flow on the Ross Ice Shelf blocked by the Transantarctic Mountains

Given:  = 80°S Undisturbed lower layer depth = 500 m Depth of lower layer adjacent to Transantarctic Mountains = 1000 m Density of bottom layer = 1.3 kg m-3 Density of upper layer = 1.2 kg m-3

How does the atmosphere respond to this pressure difference?

Rossby radius of deformation (rR) – the horizontal distance over which the atmosphere will adjust to geostrophic balance

0.5 1 ж Dq ц rR = зgh ч f и qbot ш

What is the Rossby radius of deformation for the example above?

We can estimate the horizontal pressure gradient for a blocked flow as:

¶p Dp » ¶x rR

The geostrophic wind induced by blocked flow can then be calculated as:

1 ¶p v = g r f ¶x

1 Dp v » g r f r R Barrier wind – the geostrophic wind that develops in response to a blocked flow

What is the geostrophic wind speed for the blocked flow in the example above?

What is the geostrophic wind direction?

What conditions lead to the development of barrier winds on the Ross ice shelf?

Barrier winds on the Ross Ice Shelf: 12 UTC 11 March 2005