The Thermal Wind By Sandro Lubis G24063245

Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University

Thermal Wind Sandro Wellyanto Lubis

HH

Bogor Agricultural University Department of Meteorology and Geophysic [email protected] , +62856634415/+6281385644350 HH

March 1 , 2009

Abstract One of the layer on earth which is very mysterious to be studied is atmosphere. It consist of various type of layer charactheristic and system which always move to a equilibrium point. One of the system is the wind. Discussing about the wind its very related with dynamic meteorology that is the study of the motion of the atmosphere that are associated with weather and climate. At this moment I will describe the thermal wind balance, derive the key equation and apply this concept on the movement of thermal wind prediction.

1. The Thermal Wind Equation in Geopotential Isobaric Coordinate

so:





the

balance

of

     (fVg )  (k x  P Φ) p p



geosthropic wind



The thermal wind is defined as geostrophic winds vertical derivative (since p is now independent) is:

    D VH  P Φ  f k x V H Dt

in



  1  (k x Vg )    P Φ f    1  k x(k x Vg )   k x  P Φ f  1  Vg  k x  P Φ f

We know that the momentum equation in isobaric coordinate system for frictionless large-scale quasi horizontal flow is:

that



f( k x Vg )    P Φ

Firstly, before we try to get the solution of thermal wind equation we have to find the geostrophic wind equation and the vector difference between geostrophic winds at two levels will form the thermal wind equation.

Remember



0    P Φ  f k x Vg

D VH = 0 or constant Dt

     f Vg  (k x  P Φ) p p

because a geostrophic flow is defined as a frictionless (Large Reynolds number), shallow, low Rossby number flow. Since the Rossby number is small we neglect acceleration effects.



f

 Vg p





 (k x  P

Φ ) p

Hydrostatic relation can be written as:

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

1

Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University

1 p g ρ z RdT z Φ  g  p p p 

P1

  T 

P1





Tlnp

Po P1

 lnp

Tlnp

Po

ln P1

P0

Po P1

Then, use the hydrostatic equation to complete the thermal wind equation:



P Tlnp   T  ln 1  Po

 Po



 Vg

RdT f  (k x  P  ) p p 

  



Finally we can get the Thermal equation



 Vg

p

p

 (

Rd   k x  P T) f



VT  



 Vg lnp



Rd   k xP T f



VT 

P Rd   k x  P  T  ln 1 f Po

P Rd   k x  P  T  ln 0 f P1

The equation above, in some case its difficult to be solved. So, we need the solution in simple form: 

 Vg  (

 T    T 

Rd   k x  P T ) lnp f



k



VT



Vg 1

P1



 Vg 





Vg 0 

 Po

Rd   ( k x  P T ) lnp f









V p T 



V go

V g1

P1

Vg 1  Vg 0  (  VT  

T 

Rd   k x  P )  Tlnp f Po

 T    T 

 P1

Rd  k xP f



Tlnp

Po

2. The Thermal Wind Equation in Level Coordinate (z as Vertical coordinate) The thermal wind is depend on the vertical shear of the geostrophic wind on the temperature structure of the atmosphere, so firstly lets see the motion equation for frictionless is defined as

By calculus, we can solve this form P1



Tlnp

Po





P1

  DVH 1   H p  f k x VH DT ρ

P0

  1 0   H p  f k x V g ρ

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

2

Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University

1 p 1 ρ 1 R 1 T    p z ρ z R z T z 1 ρ 1 p 1 T   ρ z p z T z 1 ρ  ρg 1 T   ρ z p T z 1 ρ  g 1 T   ρ z RT T z

  1 f(k x V g )    H p ρ   1  (k x V g )    H p ρf    1   k x(k x V g)   (k x  H p) ρf    1 V g  (k x  H p) ρf





 H p   H ( RT)

After we find the geostrophic equation, Now derive the equation with respect to z:





 H p  R  H ( T)

  Vg 1   (k x  H p) z ρf







 H p  RT  H   Rρ  H T



Vg 1   1   p   2 k x H p  k x H p z ρf z ρ f

Now combine these equation:

Remember hydrostatic relation

p =- ρg z



Vg 1 p  g    Vg  k x  H ρ z ρ z ρf

so, 

g  Vg  RT  g  VT  Vg  RT 

Vg 1   1     2 k x  H p  k x  H p(ρg) z ρf ρf  Vg

VT 



 Vg 1 p 1   1    ( )( k x H p) k x H(ρg) z ρ z ρf ρf



VT  

1 T  g  1  ρ Vg  k x(  H p   H T) T z ρf RT T    1 T g 1 Vg  ( k x  H p)  T z RT ρf

gρ   k x  H T) ρf T 

1 p  g   Vg  k x  H ρ ρ z ρf

VT 

This equation of thermal wind can be change in to another form by using the equation of state where p  RT

g  1 T  g  g   Vg  Vg  Vg  k xH T RT T z RT fT



VT 

1 T  g   Vg  k xH T T z fT

Lets change this form:

lnp  lnρ  lnR  lnT

This last equation clearly exhibits the dependence of the vertical shear of the geostrophic wind on the temperature structure of the atmosphere.

    (lnp)  (lnρ)  (lnR) (lnT) z z z z

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

3



Vg

Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University



3. The Thermal Wind Equation in Isobaric Coordinate ( p as Vertical Coordinate)

T0

VT

By modification the equation of motion to isobaric coordinate where the pressure gradient is transformed because pressure as vertical coordinate, we get new equation of motion:

T0  T 

H T Prediction of Thermal Wind Direction in the Northern Hemisphere.

    D VH   g  p z  f k x VH Dt

The temperatures in this thermal equation should actually be virtual temperatuires, but difference between actual temperature and virtual temperature is usually quite small in the free atmosphere, and we use the dry air temperature (T) in practice (Riegel, 1992).

By approaching of geostrophic wind, where the geostrophic wind is the wind which would exist if the flow were totally

By empirical observation or scaling analysis, we can see that the first terms in are negligible relative to the horizontal temperature terms so we can approximate our results as:

V



unaccelerated  H

D VH 0 Dt

thus 

 V

obtain the coordinate :

g

, and the result we will

geostrophic





VT 

and

wind



in

p-



0   g  p z  f k x VH

g   k xH T fT







f k x VH   g  p z   g k x VH    p z f    g   k x(k x VH )   (k x  p z) f  g    Vg   (k x  p z) f  g  Vg  k x  p z f

1 T  Vg T z depends largely on the static stability of the atmosphere and is usually less than In Riegel (1992), the first term

g   k x H T , fT so the first term from the equation can be neglected. 1/10 of the second term

To obtain the thermal wind equation we have to derive this equation (since p is now independent). Its vertical derive is 

Vg Colder air o

VT

T

V g z1 

V g z2  Warmer air

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

T  21 C o

22 C 23o C 24o C

4

p



g   z k xp f p



g    α k x  p    f  g



Vg p 

Vg

1    k x p α p f

Hydrostatic equation

Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University  

VT  

V g

In spherical coordinate the geostrophic balance may be written as:

p

 fv g   

VT 

1   k x p  f

fu g  

 1   x a cos  

 1   y a 



The negative sign in front of V g / p

Combining these with hydrostatic balance, ∂  /∂z=b, it will give:

arises from the fact that although pressure increases downward we are interested in the variation of geostrophic wind with increasing height (i.e. decreasing presure) (Riegel, 1992).

 f

Now, using the equation of state where p=ρRT or pα=RT so :

f

1    RT   VT  k x  p  f  p  

 1 R VT  k x  p T f p



VT 

T

vg z

ug z





b 1 b  x a cos  

b 1 b  y a 

These equations are known as the thermal wind balance and the vertical derivative of the geostrophic wind is the thermal wind.

R   k xp T fp

5. Application of Thermal Wind The thermal wind as we know is vector difference between geostrophic winds at two levels.

T  T  



VT or ( k x p T ) 

p T The Thermal wind on an isobaric surface nothern hemisphere (f>0) 4. The Thermal Wind Equation in Spherical Coordinate Thermal wind balance arises by combining the geosthropic and hydrostatic approximations, and this most easily done in context of the anelastic (or Boussinesq) equation.

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

The thickness of a layer between two isobaric surfaces is proportional to the temperature of the layer. The thermal wind is parallel to the isotherms with

5

Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University

warm air to downstream

the

right

   VT  V g 1  V g o  VT  (10iˆ  10 ˆj )m / s

facing

P Rd   k x  P  T  ln 0 f P1



VT 

287 ˆ  900 k x  P  T  ln 4 10 700  287 900 10iˆ  10 ˆj   4 kˆ x  P  T  ln 10 700  287 900 14.14  4 kˆ  p  T  sin 90 ln 700 10   p  T   1.96 K / 100 km (10iˆ  10 ˆj ) 

 1.96 o C / 100km

Directional changes of the thermal wind with height can be used tofind if warm air or cold air advection is occurring in the layer.

And the direction is   1.96 o C  1 .96 o C  P  T   Cos 225 iˆ  Sin 225 ˆj  100 km  100 km 

If the Geostrophic Wind is: Veering with height = WAA Backing with height = CAA

Or we have another way is

Problems 1: Determine the mean temperature gradient between isobaric 900 hPa and 700 hPa if the geostrophic wind at 900 hPa level is 1o m/s , northward; and the geostrophic wind at 700 hPa level is 10 m/s and westerly.



VT  



Vg o 

p T



V g1

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

1 Rd P0 ln f P1



VT

   1    k x k x  P  T    k x VT   Rd ln P0 f P1

VT 2250



k x  P  T 

Answer:



Rd   P k x  P  T  ln 0 f P1

6

Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University

At 844 hpa (1524m)winds are 28 kts at 315 degrees   1       k x VT  k x  P  T   k   k x k   P  T  Rd P0     ln f P1

1



  P  T 



Rd P0 ln f P1

VT 



k x VT

1



so,  P  T  

To solve this problems we can use this equation:

Rd P0 ln f P1

u ^ v ^ i j z z

And Then the thermal wind equation in vector form is 



k x VT

f VT  

g T T

Breaking up into vector components:

Problems 2: ^

^

u878  11.47kts i  16.4kts j

Given the following Sounding data determine temperature gradient direction at 850 hpa (1476 m):

^

^

u 844  19.8kts i  19.8kts j

 19.8  11.47kts ^  19.8  16.4kts V T   i  1524  1219 m 1524  1219 m ^ ^     0.014 i  0.0057 j s 1  

T   f  VT

 0.514m / s j   1kts

^

T   f  VT g

^ ^ ^        1  10 4 s 1 k    0.014 i  0.0057 j s 1    

^  T    5.7 10 7 s 2 i  1.4 10 6 s  2 

 j 

We see a negative gradient in the east west direction and a negative gradient in the north south direction.

Answer: at 878 hpa (1219 m) winds are 20 kts at 325 degrees

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

^

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Dynamic Meteorology, Department of Applied Meteorology Bogor Agricultural University

About Writer

u (878hPa) 1219m u (844hPa) 1524m

North-south

u g 844hpa  u g 878hpa 

Cold

T

East-west

Warm

Name Date of Birth Major Minor My Blog

Bibliography

[Anonim]. 2009. Oceanography. http:// www .usna. edu. [Bogor, 2 Maret 2009].

Bachelor Programme of Apllied Meteorology, Department of Meteorology and Geophysic, Bogor Agricultural University, Indonesia.

Holton, James R. 2004. An Introduction to Dynamic Meteorology 4th Edition. Elsevier Inc

“Its simple writing, so i do need additional information about this theme to make it completly and perfect, my aim is just to derive equations of thermal wind in various type of coordinate system  ”

Riegel,C. A. 1992. Fundamentals of Atmospheric Dynamic and Thermodynamics. USA: World Scientific Publishing Co. Pte. Ltd. Vallis, Geoffrey K. 2005. Atmospheric and Ocean Fluid Dynamic. USA: Cambridge University Press.

Sandro Wellyanto Lubis, Thermal Wind Bachelor Programme of Applied Meteorology

: Sandro Wellyanto Lubis : Sabang, June 15 1988 : Apllied Meteorology : Marine Science : www.sandrolubis.wordpress.com

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