6.7 Thermal wind in pressure coordinates

Transcription

1 176 CHAPTER 6. THE EQUATIONS OF FLUID MOTION 6.7 Thermal wind in ressure coordinates The thermal wind relation aroriate to the atmoshere is untidy when exressed with height as a vertical coordinate (because of ρ variations), but becomes simle when exressed in ressure coordinates.. To roceed in coordinates, we write the hydrostatic relation thus: z = 1 gρ. and take, for examle, the -derivative of the x-comonent of (6.58) yielding = g 2 z f y = g µ z = 1 µ 1. f y f y ρ Since 1/ρ = RT/, itsderivativeat constant ressure is µ 1 = R µ T, y ρ y whence = R µ T. (6.62) f y Similarly, for v we find v = R µ T. (6.63) f x Thus, horizontal gradients of temerature must be accomanied by vertical gradients of wind. (In height coordinates, one can obtain a similar relationshi, but it is a bit messier, because of the ρ factors in (6.57).) Eqs. (6.62) and (6.63) exress the thermal wind relationshi in ressure coordinates. To see how, consider Fig.(6.40). Suose, for simlicity, = 0 is constant at sea level (z =0), and that there is a monotonic decrease of temerature with y (so T/ y < 0) within the atmoshere. Then geostrohic balance tells us that u =0at z =0. Now, hydrostatic balance tells us that, at oints α and β aloft, the ressure is (x, y, z) = 0 g Z z 0 ρdz.

2 6.7. THERMAL WIND IN PRESSURE COORDINATES 177 Figure 6.40: Since beneath oint α (on the left) the air is warm and therefore light (low density), while is cold and dense beneath oint β, it follows that α > β. Aloft, therefore, there must be a geostrohic flow, out of the aer and therefore westerly (if we are in the northern hemishere), with low ressure on its left. Thus, a northward decrease of temerature imlies westerly winds increasing with height (i.e., / < 0), consistent with (6.62). In articular, if temerature decreases oleward (as we have seen it does), then T/ y < 0 in the northern hemishere, T/ y > 0 in the southern hemishere, so f 1 T/ y < 0 in both. Then (6.62) tells us that / < 0: so, with increasing height (decreasing ressure), winds must become increasingly eastward (westerly) in both hemisheres (see Fig.??) which is just what we observe in Fig Thermal wind exressed in terms of otential temerature The thermal wind can also be written in terms of otential temerature, θ, defined in eq(4.7) of chater 4. From eq(4.7) we note that: µ µ k µ T θ = y o y and so, for examle, eq(6.62) can be exressed thus:

3 178 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.41: = R T f θ µ θ = 1 y fρθ µ θ. (6.64) y (and similarly for v ) where the equation of state has been used. Thus the vertical (geostrohic) wind shear in ressure coordinates is directly roortional to the meridional gradient of θ on ressure surfaces. The connection between meridional otential temerature gradients and vertical wind shear is readily seen in the zonal-average climatology - see, for examle, fig.5.16 and fig.5.6. Be sure to check out that eq(6.64) also makes quantitative sense. The atmoshere is also close to thermal wind balance on the large scale at any instant. Fig.(6.42) lots the otential temerature on the 500mb surface on Nov 6th, 2001, zero hours GMT. Note the strong meridional gradients in middle latitudes associated with the jet stream. These gradients are also evident in fig.(6.43), a vertical cross section of otential temerature, θ, and zonal wind, u, through the atmoshere at 90 Wextendingfrom20 Nto70 N at the same time as in fig(6.42). The vertical coordinate is ressure. Note that, in accord with eq(6.64), the wind increases with height where θ surfaces sloe uward and decreases with height where θ surfaces sloe downwards. The vertical wind shear is very strong in regions where the θ surfaces steely sloe; the vertical wind shear is very weak where the θ surfaces are almost horizontal.

4 6.7. THERMAL WIND IN PRESSURE COORDINATES 179 Figure 6.42: The otential temerature, θ, on the 500mb surface on 6th November 2001 at 00 GMT.

5 180 CHAPTER 6. THE EQUATIONS OF FLUID MOTION Figure 6.43: A cross section of zonal wind, u, and otential temerature, θ, through the atmoshere at 90 W extending from 20 Nto70 Non6th November 2001 at 00 GMT.

6 6.7. THERMAL WIND IN PRESSURE COORDINATES 181 Summary of key equations 9 (x, y, z) ³ coordinates (x, y, z) ³ coordinates (x, y, ) ³ coordinates,,,, x y z x y z,, x y (incomressible - OCEAN) (com. erfect gas - ATMOS) Continuity: ρ t u =0 u =0 Hydrostatic balance z = gρ z = gρ z = 1 gρ Geostrohic balance fu = 1bz ρ fu = 1bz ρ fu =gbz z Thermal wind balance f z = gbz ρ f = R bz T 9 Note that (x, y, ) is not a right-handed coordinate system. So while bz is a unit vector oint toward increasing z, and therefore uward, bz is a unit vector oint toward decreasing and therefore also uward.

7 182 CHAPTER 6. THE EQUATIONS OF FLUID MOTION