Weather and Climate Laboratory Spring 2009

Transcription

1 MIT OenCourseWare htt://ocw.mit.edu Weather and Climate Laboratory Sring 2009 For information about citing these materials or our Terms of Use, visit: htt://ocw.mit.edu/terms.

2 Thermal wind John Marshall, Alan Plumb and Lodovica Illari March 4, 2003 Abstract We describe thermal wind balance, derive key equations and discuss the underlying hysics. 1 1 The thermal wind equation Thermal wind is the most fundamental and significant dynamical balance controlling the large-scale circulation of the atmoshere and ocean. It is a consequence of hydrostatic and geostrohic balance, and relates horizontal buoyancy gradients to changes in the horizontal wind with height. If the flow is such that the Rossby number is small R o 1 where U R o = (1) fl (here U is a tyical horizontal current seed, f is the Coriolis arameter and L is a tyical horizontal scale over which U varies), then the Coriolis force is balanced by the ressure gradient force in the horizontal comonent of the momentum equation and the flow is close to geostrohic balance: or, writing out its Cartesian comonents, u g = 1 bz, (2) ρf u g = 1 ρf v g = 1. ρf x y ; (3) If we now combine Eq.(2) with hydrostatic balance, by eliminating the ressure term, we arrive at the thermal wind equation. We now go through this rocedure first for an 1

3 1 THE THERMAL WIND EQUATION 2 incomressible fluid, such as water, and then for a comressible fluid like the atmoshere to derive various forms of the thermal wind equation. 1.1 Thermal wind equation for an incomressible fluid such as water Suose that the density of water varies thus: δρ ρ = ρ + δρ and << 1 o where ρ o is a constant reference density, and δρ is the variation of the density about this reference. Note that this is a very good aroximation because, tyically, the density of the water in our rotating tank exeriments varies about its reference value by less than a fraction of 1%. Now take / of Eq.(3) (relacing ρ by ρ o where it aears in the denominator) we obtain, making use of the hydrostatic relation + ρg = 0,: ρ o g ρ = ; fρ o y v g g ρ = fρo x (4) or, in vector notation g = bz ρ. (5) fρ o So if ρ varies in the horizontal then the geostrohic current will vary in the vertical. To exress things in terms of temerature, and hence derive a connection between the current/wind and the thermal field (called thermal wind) wenotethatthe densityofwater, to an aroximation which is useful, deends on temerature T in a linear fashion: ρ = ρ o (1 α (T T o )) (6) where α is the exansion coefficient of water at T = T o,and T o is a reference temerature. Thus (5) can be written: αg = bz T (7) f This is a simle form of the THERMAL WIND relation. It tells us nothing more than the hydrostatic and geostrohic balance, but it exresses these balances in a different way. We see that there is an exactly analogous relationshi between ug and T as between u g

4 1 THE THERMAL WIND EQUATION 3 and : comare Eqs.(7) and (2). So if we have horizontal gradients of temerature then the geostrohic flow will vary with height. 1.2 Thermal wind in ressure coordinates Eqs.(4) ertain to an incomressible fluid such as the water or the ocean. The thermal wind relation aroriate to the atmoshere can be written down but it is untidy when exressed with height as a vertical coordinate (because of ρ variations). However it becomes simle when exressed in ressure coordinates. To roceed in coordinates, we write the hydrostatic relation thus: 1 =. gρ and take, for examle, the ³ -derivative of the x-comonent of the geostrohic wind in ressure g g coordinates (u g,v g )=, yielding: f y f x 2 z g µ 1 µ 1 = = =. f y f y f y ρ Since 1/ρ = RT/, its derivative at constant ressure is µ µ 1 R T =, y ρ y whence µ u R T =. (8) f y Similarly, for v we find µ v R T =. (9) f x Eqs.(8) and (9) exress the thermal wind relationshi in ressure coordinates. In height coordinates, one can obtain a similar relationshi, but it is less elegant because of the ρ factors in Eq.(3). To see how horizontal gradients of temerature must be accomanied by vertical gradients of wind in the atmoshere, consider Fig.(1). 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 =0 at z =0. Now,

5 1 THE THERMAL WIND EQUATION 4 Figure 1: A schematic diagram illustrating why horizontal gradients of temerature must be accomanied by vertical gradients of wind in the atmoshere. 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. Since beneath oint A (on the left) the air is warm and therefore light (low density), while is cold and dense beneath oint B, it follows from hydrostatic balance that A > B. 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., u/ < 0). hydrostatic balance tells us that, at oints A and B aloft, the ressure is (x, y, z) = 0 g Z z 0 ρdz. Since beneath oint A (on the left) the air is warm and therefore light (low density), while is cold and dense beneath oint B, it follows that A > B. 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., u/ < 0), consistent with Eq.(8). 1 Notes to accomany : Weather and Climate Laboratory. For a more detailed descrition see notes on web age here: htt://aoc.mit.edu/labweb/notes.htm