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 fv = 1 ρ x (1) dv dt + fu = 1 ρ y, (2) where f = 2Ω sin φ. The two largest terms are the Coriolis and ressure gradient forces (PGF) which combined reresent geostrohic balance. We can define geostrohic winds u g and v g that exactly satisfy geostrohic balance, as fv g 1 ρ x (3) fu g 1 ρ y, (4) and thus we can also write du dt = f(v v g ) (5) dv dt = f(u u g ). (6) In other words, on the synotic scale, accelerations result from deartures from geostrohic balance. z R Q δx δz +δ x Figure 1: Isobaric coordinates. It is convenient to shift into an isobaric coordinate system, relacing height z by ressure. Remember that ressure gradients on constant height surfaces are height gradients on constant ressure surfaces (such as the 500 mb chart). In Fig. 1, the oints Q and R reside on the same isobar. Ignoring the y direction for simlicity, then = (x, z) and the chain rule says δ = δx + x z δz. 1
Here, since the oints reside on the same isobar, δ = 0. Therefore, we can rearrange the remainder to find δz δx = x. Using the hydrostatic equation on the denominator of the RHS, and cleaning u the notation, we find 1 z = g ρ x x. (7) In other words, we have related the PGF (er unit mass) on constant height surfaces to a height gradient force (again er unit mass) on constant ressure surfaces. We will ersist in calling this height gradient force a PGF or, more secifically, the isobaric PGF. Recalling geootential dφ = gdz, we can also get 1 ρ x = Φ x. (8) Similarly, z 1 ρ y = Φ y. (9) Note density does not aear in the isobaric PGF. This is the rincial advantage of isobaric coordinates. Further, we can write an equation like (8) as 1 ( ) Φ ρ x = x to remind ourselves that the height gradients are comuted on isobaric surfaces. Therefore, the geostrohic wind equations on isobaric surfaces are u g = 1 ( ) Φ f y v g = 1 ( ) Φ f x The hydrostatic equation in isobaric coordinates is (10) (11). (12) dφ d = RT. (13) The thermal wind Consider the familar situation shown in Fig. 2, in which the oles are colder than the equator. Suose isobaric surface tilts down towards the north. On a constant ressure surface, ressure value resides closer to the surface, and reresents a locale of lower geootential height. Figure 3 deicts this situation. PGF oints towards lower geootential heights, Coriolis acts to the right following the motion (in the Northern Hemishere), and thus the geostrohic wind blows arallel to isoheights with lower height to the left. It is a westerly wind in this examle. 2
z -2δ z2 -δ z1 Pole (cold) Equator (warm) Figure 2: Pole colder than equator. Φ δφ PGF geostrohic wind Coriolis Φ Figure 3: Geostrohic wind on isobaric surface. Also note thickness z1 < z2, since the average layer temerature of the former is lower. As a consequence, isobaric surfaces located farther aloft will sloe rogressively more strongly down towards the ole with height. This means the geootential height gradient also increases, making the geostrohic wind stronger. (Note we don t have to consider density anymore; that s already imlicitly factored in.) Thus, not only does the geostrohic wind continue blowing from the west, there is also a westerly vertical wind shear. Choosing a latitude residing somewhere in between ole and equator, this simle examle gives us a vertical wind rofile like this: z, - x Figure 4: Westerly vertical shear of the geostrohic wind. 3
Therefore,we can relate the vertical shear of the zonal (west-east) geostrohic wind to the meridional (north-south) temerature gradient. Similarly, temerature gradients in the east-west direction imly vertical shear of the northerly comonent of the geostrohic wind. I am sticking with the north-south T scenario merely because it s slightly easier to icture. The thermal wind refers to the vertical shear of the geostrohic wind. In ressure coordinates, the vertical shear is and v g. Since ressure decreases with height, differentiate it with resect to ressure. Thus < 0 means u g increases with height. Take (11) and = 1 f [ ( ) ] Φ. y Interchange the order of differentiation on the RHS and use the isobaric hydrostatic equation (13) and find, after further rearrangement, ln = R ( ) T. (14) f y That is, the vertical shear of the westerly geostrohic wind deends on the north-south T. For the northerly geostrohic wind, we would find ( ) v g T ln = R. (15) f x If we define the vector geostrohic wind as V g = u g î + v g ĵ, then V g ln = R f ˆk T, (16) where reminds us to comute the temerature gradient on an isobaric surface. This is the thermal wind equation. It is NOT A WIND. It is a wind shear. Also, it does not involve the true wind, but rather the geostrohic wind. If the wind is not geostrohic, then the thermal wind equation is not exact, and may even mislead. Finally, it relates geostrohic shear to temerature gradients on isobaric surfaces not constant height surfaces. By the nature of the cross roduct, (15) also shows the vertical shear is arallel to isotherms, since it must be orthogonal to T. We can see this more easily if we actually define a vertical shear vector V T (where the subscrit T stands for thermal ) by integrating between two isobaric surfaces 0 and 1 1 VT V = 1 g ln d ln = R ˆk T, f 0 0 The LHS is simly V T = V g ( 1 ) V g ( 0 ) u T î + v T ĵ, where u T and v T are the shear vector s comonents. The RHS is messy, but simlifies a lot if we take a layer mean T between the two ressure levels. So, the thermal wind shear comonent equations are u T = R [ ] T ln 0 (17) f y 1 v T = R f [ T x 4 ] ln 0 1. (18)
The hysometric equation ermits us to rewrite the RHS of the above as u T = 1 f y [Φ 1 Φ 0 ] (19) v T = 1 f x [Φ 1 Φ 0 ]. (20) There are three elements in the receding. For the 0 to 1 layer, there is the geostrohic wind at the layer bottom V g ( 0 ), the geostrohic wind at the layer to V g ( 1 ), and the vertical shear V T, itself determined by the horizontal gradient of layer mean temerature. If we know any two of these, we can get the third. Kee in mind that since u T is roortional to T y and v T is roortional to T x that V T is arallel to isotherms in the 0 to 1 layer. 5