The thermal wind 1. v g

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Jonathan Butler
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1 The thermal win The thermal win Introuction The geostrohic win is etermine by the graient of the isobars (on a horizontal surface) or isohyses (on a ressure surface). On a ressure surface the graient of the isohyses reflects the tilt of the ressure surface. If this tilt changes with ressure then also the geostrohic win will change with ressure in magnitue an/or irection. Generally seaking the thermal win is the change of the geostrohic win with ressure (or height): it is the vector ifference of the geostrohic win at two ifferent levels an as such it is not a real win. In this note we will treat the causes of the thermal win, an we will also see how this quantity can be use in ractice. The geostrohic win The exressions for the geostrohic win are: u g = ρg y an v g = ρ g x, () where the minus sign in the exression of the left is only imortant for the irection of the u-comonent of the geostrohic win. On a ressure surface with isohyses these exressions may be rewritten assuming hyrostatic equilibrium: This leas to: = ρg, (2) z u g g z = an f y v g g z = (3) f x From formula (3) it aears that the magnitue of the geostrohic win eens on the tilt of the ressure surface ( z/ y an z/ x). For a given istance ( y or x) of the isohyses the magnitue of the geostrohic win can be etermine with the ai of Equation 3. In the examle in Figure it is obvious that Μz/Μx < 0 (height z of the ressure surface ecreases in the ositive x-irection) an consequently for the v- comonent of the geostrohic win we have: v g < 0. The larger the tilt of the ressure surface the smaller the istance ( x) between the isohyses an hence the larger the geostrohic win see. A similar conclusion alies to the u-comonent of the geostrohic win. It aears that the magnitue (an the irection) of the geostrohic win may change if ressure surfaces are NOT arallel to one another. Just when such a situation may occur is exlaine in the next section.

2 The thermal win 2 Figure. The magnitue of the geostrohic win is etermine by the tilt of the ressure surfaces. For a small tilt (left) the isohyses are far aart an the magnitue of the geostrohic win is small; for a large tilt (right) the isohyses are close together an the magnitue of the geostrohic win is large. The relation with temerature Accoring to the equation of state (i.e. the gas law) the temerature etermines the air ensity (ρ) for a given value of the ressure (): = ρrt hence ρ = (4) RT Thus in an area with high temeratures the ensity at a given ressure is lower than in an area with low temeratures (where the ensity is higher). Because of the high ensity the ecrease of ressure with height in the col area is larger than the ecrease of ressure in the warm area. In Figure 2 (left) this is clearly visible. We have assume that the ressure at the surface (z = 0) is the same everywhere ( 0 ). In the col area less vertical istance is neee to have the same ecrease in ressure (). Figure 2. The effect of a horizontal temerature ifference (left) an a horizontal temerature graient (right) on the height an tilt of a ressure surface (assuming equal surface ressure ( 0 ). If temerature changes in the horizontal, then the ressure surface aloft is no longer horizontal but will have a tilt (Figure 2 right). A horizontal temerature graient

3 The thermal win 3 influences the tilt of all ressure surfaces. From Figure 3 it aears that the tilt of the ressure surfaces increases higher u in the atmoshere. This will have consequences on the magnitue of the geostrohic win: in the case sketche in Figure 3 it will increase with height. From Figures an 3 the following conclusion can be rawn: A horizontal temerature graient causes a change of the geostrohic win with height (or with ressure). The ifference between the geostrohic win at the two ressure surfaces is calle the thermal win. Figure 3. The effect of a horizontal temerature graient on the tilt of subsequent ressure surfaces. The magnitue of the thermal win thus eens on the value of the horizontal temerature graient. An exression for the magnitue of the thermal win is etermine by ifferentiating the exression of the geostrohic win with relation to ressure (): For the factor g z g z g z u = = = g (5) f y f y f y z we use the assumtion of hyrostatic equilibrium: z = ρg = g RT. Here we have use the gas law. Substituting this in Equation (5) an multilying by leas to: ug ug = = ln R f y. (6) We integrate this exression from 0 to, with 0 > :

4 The thermal win 4 0 R u g = ln. f y 0 This leas to an exression for the comonents of the thermal win (u T, v T ): u T R = ug( ) ug( 0) = ln( 0 ) (7) f y an v T R = vg( ) vg( 0) = ln( 0 ), (8) f x In the above integration we have use an average value (T ) for the temerature in the layer 0 - in orer to simlify the integration. From these exressions it is obvious that the thermal win eens on the horizontal temerature graient. The following rule also follows from these equations: The thermal win blows arallel to the isotherms with the col air on the left han sie. The closer the isotherms the stronger the thermal win. Avection of warm an col air From Equations (7) an (8) it aears that the irection of the thermal win is arallel to the isotherms. This leas to the following two ossible situations. Figure 4. Win backing with height is an inication of col air avection. Figure 5. Win veering with height is an inication of warm air avection. In Figure 4 the win is backing going from 0 (lower level) to < 0 (higher level), in this case from west to southwest. The thermal win is the win vector on the high level minus the win vector on the low level. The thermal win blows arallel to the isotherms with the col air on the left han sie. The geostrohic win on both levels is blowing from the col area: this is a case of col air avection. In Figure 5 the win is veering from west northwest. The thermal win again has col air on its left han sie, an in this case it aears that the win on both ressure levels is blowing from the warm area: this is a case of warm air avection.

5 The thermal win 5 The value of the temerature avection is etermine by the value of the thermal win (i.e. by the istance of the isotherms, i.e. by the horizontal temerature graient) an by the comonent of the geostrohic win which is at right angles to the isotherms. From Figure 6 it aears the area of the triangle with sies v g ( 0 ), v g ( ) an v T must be roortional to the value of the avection: the larger the area of the triangle the larger the temerature avection. Figure 6. The area of the triangle is roortional to the value of the avection. Qualitative alication: hoograh an stability Avection of air from a ifferent average temerature in a number layers in the vertical influences the stability of the atmoshere. It is relatively easy to gain an overview of the avection in ifferent layers by erforming what is calle a hoograh analysis. This involves the construction of a raial iagram with the station in the centre. From this centre the win see at several or all ressure levels is lotte as a vector (Figure 7). The vector is lotte in the irection of the win. This means that a westerly win is lotte as an arrow to the east (Figure 7). Going to the next higher ressure level the next arrow is lotte etc. The en oints of all vectors are connecte by straight lines (or arrows). These straight lines are the vectors of the thermal win (ref. Figure 6). The en result is calle a hoograh. The next ste is to consier the veering or backing of the win, starting from the lowest level to ever higher ressure levels. In this way the warm or col air avection in each layer can be etermine. By comaring the areas of the triangles in relation to warm or col air avection it is ossible to etermine the changes in atmosheric stability. As an examle in Figure 7 between 70 an 50 kpa there is more col air avection than between 90 an 70 kpa. This means that in this case the vertical temerature graient must increase an instability is increasing.

The thermal win 6 Figure 7. Examle of a hoograh. Win vectors on all ressure levels (black, ressure in kpa) are all lotte from the centre, thermal win vectors (re) connect the black arrows.

6 The thermal win 6 Figure 7. Examle of a hoograh. Win vectors on all ressure levels (black, ressure in kpa) are all lotte from the centre, thermal win vectors (re) connect the black arrows. In Figure 8 a number of examles of arts of hoograhs are resente an the following conclusions about the increase of ecrease of stability can be rawn: Figure 8. Examles of the alication of hoograh analysis for etermining the change in stability of a number of layers (see text). The layers are inicate by increasing numbers from bottom to to. A: This is the case where the thermal win is equal in each layer. The areas of the three triangles are therefore equal. It means that: at every level the same amount of col air avection is taking lace. The temerature will ro everywhere, but the vertical temerature graient an hence the stability will not change. B: In this case the thermal win, the areas of the triangles an hence the col air avection increase from bottom to to. As a result this art of the atmoshere will become more unstable (or less stable) because temerature ecrease near the to will be stronger than near the bottom.

7 The thermal win 7 C: Each layer shows col air avection, but in the to layer this col air avection is larger than in the mile layer: above level the instability will increase. In the bottom layer the col air avection is larger than in the mile layer: below level 2 the instability will ecrease. D: This is a case of warm air avection. Above level the instability will ecrease because temeratures in the to layer will rise more than temeratures in the mile layer. Below level 2 the instability will increase because the temeratures in the mile layer will rise less than temeratures in the bottom layer. In ractice hoograh analysis will inicate if, at that moment, the stability on a certain location will increase or ecrease. Changes in the environmental lase rate have irect consequences for the stability an e.g. may be imortant for the eveloment of convective clous an showers. Using hoograh analysis it may be ossible to etermine if a warm front is aroaching or if a col front has asse. Quantitative alication: etermination of change in temerature From the thermal win equations (7) an (8) the value of the thermal win comonents can be calculate for a given horizontal temerature graient. If we rewrite this equation, then the value of the horizontal temerature graient can be calculate for given values of the thermal win comonents. If in an air layer the value of the comonent of the win which is at right angles to the isotherms (V N ) has been etermine (Figure 6), then the change in temerature in that layer can be calculate with the following exression: t = V n N = V N V T f R ln( 0 ) (9) This calculation assumes that no other rocesses such as vertical motions an iabatic rocesses are acting at the same time, as these may also effect the change in temerature.

ATM The thermal wind Fall, 2016 Fovell

ATM The thermal wind Fall, 2016 Fovell ATM 316 - The thermal wind Fall, 2016 Fovell Reca and isobaric coordinates We have seen that for the synotic time and sace scales, the three leading terms in the horizontal equations of motion are du dt