# physics. curtin. edu. au/teaching/units/2003/Avp201/

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Lec03.ppt Using the Equation of Motion

Objectives Revision of last weeks lecture
Applying Equation of motion to derive wind models “Anomalous” wind flows Impact of friction on air flow Thermal wind

Revision Last week we obtained a complete equation of motion which incorporated both “real” and “apparent” forces. Stated that the total rate of change of velocity with time (acceleration) was due to a combination of Pressure Gradient Force, Gravity, Centrifugal Force, Coriolis Force and Friction.

Revision We simplified matters further by combining Gravity and Centrifugal force into a single Gravity force. We also eliminated friction by assuming flow in a friction free environment, e.g. 3000ft above the Earth’s surface.

Revision By resolving into the different components of a 3-dimensional system and using scale analysis we obtained the general equation of motion as below.

Hydrostatic Equation The final term of the equation can be further simplified to give us the following result;

Hydrostatic Equation This equation tells us that as gravity is a constant, then the rate of change of pressure with height is greater for cold dense air than for warm less dense air. We can say therefore that the rate of change of pressure with height is dependent on temperature.

Use of Hydrostatic Equation
Main use is in measurement of height above ground If a ‘standard’ atmosphere is assumed whereby mean sea level temperature is 15°C and the lapse rate is 6.5°C/km, then a ‘standard’ distribution of pressure with height results This ‘standard’ is used in pressure altimeters, which sense pressure but read out height.

Wind Equations The horizontal components of Newton’s 2nd law are sometimes called the wind equations. For both N-S and E-W flow the only forces we need consider are the Pressure gradient force, the Coriolis force and Friction. We can neglect friction if we assume flow in a friction free environment, i.e. above 3000ft.

Wind Equations We have also seen from our scale analysis that we have an acceleration which is an order of magnitude less than the forces which cause it. Therefore we can disregard these accelerations, and if we have no local curvature effects such as those found around lows and highs we can state the following.

Geostrophic Wind Geostrophic motion occurs when there is an exact balance between the HPGF and the Cof, and the air is moving under the the action of these two forces only. It implies No acceleration eg Straight, parallel isobars No other forces eg friction No vertical motion eg no pressure changes

Geostrophic Wind As geostrophic conditions imply no acceleration or friction, we can set these terms to zero in the simplified equations of motion to get the Geostrophic wind equations

Geostrophic Wind We can combine these results to give an equation for the geostrophic wind on a surface chart if we know the perpendicular distance n between isobars. The equation is as follows;

An Example What is the Geostrophic wind speed for a pressure gradient of 2hPa/Km and density of 1.2kgm-3 at a latitude of 20° ? ( = x 10-5 ,2  = 1.45x10-4)

The Nature of Vg Geostrophic wind acts Parallel to the Isobars
If you have your back to the wind then Low Pressure is on your RIGHT 1016hPa Co Vg PGF 1012hPa

Upper level charts It can be shown that the geostrophic argument works for upper level charts as well as for surface charts. The equation for geostrophic wind at upper levels loses the density term and becomes;

Gradient Wind Vgr Wind which results when the Centrifugal Force resulting from curved flow is exactly balanced by the Coriolis and Pressure Gradient Forces Ce= Co- PGF 3 Cases of Vgr exist Anti-clockwise flow (a High) Clockwise flow (A Low) Straight Flow (Vgr=Vg which is a special case)

The equations for the gradient wind depend on whether the flow is cyclonic or anti-cyclonic, but it can be shown they are as follows;

Gradient Wind There are some limiting factors to gradient flow around high pressure systems when we look at the equation closely.

Gradient Wind This tells us that there is a limit to how fast the wind can move around an anti-cyclone, and that limit is twice the speed of the Geostrophic wind. There is no limit to the speed a cyclonic circulation can achieve.

Gradient Wind This tells us that when the radius of curvature is small, then so must the rate of change of pressure with distance. In other words the isobars must get further apart the closer you get towards the centre of the anticyclone. There is no limit to the spacing of the isobars around the centre of cyclonic flow.

Gradient Wind The previous slide shows us the balance of forces required to make Gradient flow occur. From our knowledge we can now say that gradient flow around a cyclone is sub-geostrophic, and that gradient flow around an anti-cyclone is super-geostrophic.

Cof Pgf Cef Wind direction In this situation, it is impossible to achieve balanced flow, as all the forces are acting in the same direction. Therefore it is impossible for clockwise flow to exist around a high in the Southern Hemisphere.

Cyclostrophic Flow As mentioned previously there are no restrictions to the strength of the pressure gradient around low pressure systems. This can lead to situations whereby if the radius of curvature is very small (such as found around tornadoes), then the centrifugal force and pressure gradient forces balance each other. This is Cyclostrophic flow.

Cef Wind direction Pgf Cof In order for balanced flow to occur, the Cef must balance the Cof and PGF. This can only happen with large amounts of Cef, eg small radius of curvature and large speeds. Therefore can only occur with small scale systems such as dust devils and tornadoes.

Friction So far we have chosen to ignore the effects that friction has on air in motion, by looking at motion above 3000ft. However, we have to take it into account when looking at motion closer to the surface.

Frictional Effects Wind veers as Coriolis is reduced
Frictional effects reduce Wind speed Cross Isobar flow towards Low Pressure Region Flow outwards from High Pressure Flow inwards to Low Pressure As friction reduces with height, wind flow will BACK with height

Frictional Effects 1012 Vgr (3000ft) Sfc wind 1010
Friction at Surface is greatest and we get a reduction in speed, which in turn leads to a reduction in Cof. PGF becomes dominant and so wind blows towards LP. Vgr = W’ly SFC = NW’ly From Vgr to SFC, winds have veered. From SFC to Vgr, winds have backed.

Effects of Friction Balanced Gradient flow
High Low Vgr Ce PGF Cof

Effects of Friction Low Vf High
Ce PGF Cof V Vf F Friction [F] reduces gradient Speed [V] Cof now reduced PGF becomes dominant force and so wind VEERS

Effect of Friction Cross - Isobar Flow
H L Co PGF Wind speed reduced by friction and so Co decreases. PGF>Co

Friction Effects Cross-Isobar Flow

Diurnal Variation of Wind
Wind shows a marked diurnal variation Peak during daytime and lull overnight Variation more marked with existence of low level temperature inversion. During night inversion acts as “lid” and prevents energy transfer downwards and thus have (relatively) stronger winds above inversion. During day convective currents break inversion down allowing sfc winds to increase and transfer turbulence aloft to decrease upper winds.

Diurnal Variation of wind

Terrain Induced Turbulence
Varying terrain will cause changes in the depth of the turbulence Be aware of changes in turbulence depth with changes in surface features i.e. Sea surfaces will have a lesser depth of turbulence than land surfaces

Wind Shear The variation of wind between 2 points
Vertical wind shear is the variation in the wind between 2 layers In particular it is the Wind at top of layer minus wind at bottom of layer

Wind Shear 270/60 10000 FT Wind Shear 270/25 5000 FT
= 270/35

Thermal Wind This shear is given the name Thermal wind.
Use of the Hydrostatic assumption and gradient wind equation can show us how the vertical shear will vary due to horizontal temperature gradients.

A B WARM COLD P = 1013 hPa Equator Pole Two columns of air, A and B exert the same pressure (1013hPa) at the surface

A B WARM COLD Constant Height AGL Equator Pole However, because A is warmer and therefore less dense than B, the pressure at the constant height surface at A is greater than at B.(By using the hydrostatic assumption and remembering that pressure drops more rapidly in cold air than warm air)

Pgf A B WARM COLD Constant Height AGL Equator Pole This sets up a Pressure Gradient Force between A and B

Co Pgf A B WARM COLD Constant Height AGL Equator Pole Coriolis force now acts on the moving air to deflect it to the left in the Southern Hemisphere to achieve balanced Geostrophic flow

Resultant Wind A B WARM COLD Constant Height AGL Equator Pole
This shows us that upper level westerlies are a direct result of horizontal temperature differences. We can now go one step further and show that we can determine the direction and strength of upper level winds if we know the direction and strength of the lower level wind and the direction and strength of the thermal wind. [Remember that upper-lower = thermal lower + thermal = upper] Giving us the upper level westerly which we observe as being the pre-dominant wind in the upper atmosphere

Web References www.met.tamu.edu/teach.html