Balance. in the vertical too

Balance. in the vertical too

Gradient wind balance f n Balanced flow (no friction) More complicated (3- way balance), however, better approximation than geostrophic (as allows for centrifugal acceleration due to curvature of parcel trajectory) Solving for wind speed (plug and chug in the quadratic formula): f f n defined as positive, and must be real, thus four possible flow configurations

Gradient versus geostrophic f n f g n f f g g 1 f Differ in magnitude by the ossby number (o= U/fL = /f) Geostrophic flow stronger for cyclonic flow (f > 0) Geostrophic flow is weaker for anticyclonic flow (f < 0) ecall from scaling arguments that o ~ 0.1 for large scale motions Geostrophic wind is within about 10% of gradient wind 3

Gradient wind solutions f f real if n f n f g n positive if f f f f n f g >0 or < 0 Thus possible physical configurations: > 0, g > 0 normal low < 0, g > 0 normal high ( > -f/) < 0, g > 0 anomalous high ( < -f/) < 0, g < 0 anomalous low ( g < 0!) Unphysical solutions if all forces in same direction. < 0 is cyclonic, > 0 anticyclonic

Possible gradient flow regular anomalous Unstable as large centrifugal required, rarely sustained Can occur near tropics (regular high crossing equator) Notice centrifugal force always out 5

Strength of highs and lows f f That is, a maximum wind speed allowed ( = g )! For a cyclone there is no such limit (as root is always positive for > 0) f f 1 For an anticyclone, < 0, real roots require f g 1 g f f g f g So, low pressure systems can be deeper than high pressure systems can be high. Highs tend to be wide and flat features Lows tend to be compact circular features 6

Gradient wind f n has possible physical solutions: > 0, g > 0 < 0, g > 0 < 0, g > 0 normal low normal high ( > -f/) anomalous high ( < -f/) < 0, g < 0 anomalous low ( g < 0!) 7

Comparison of balanced flows small scale or rapid flow large scale or slow flow ossby number Fleagle and Businger, An introduction to Atmospheric Physics, pg. 16 8

Solved! Many types of flows are balanced (speed not changing), but which forces are balancing change characteristics. Highs and lows are close to geostrophic balance, while o ~ 0.1 Lows are better described by Gradient Wind approximation (o ~ 0.1-1, and have no maximum wind speed. Highs have lower o, and have a maximum wind speed because of centrifugal forces. Implies generally weaker gradients compared to lows. Special thought experiment: Do toilets flush the other direction in the SH? Apply scaling to prove how important rotating of earth is for water draining down a plug hole. What about the vertical? What about prediction? (move beyond diagnostic relationships)

Thermal wind balance For zonal mean, we can write u g z fh T y As before, vertical (geostrophic) wind shear related to horizontal temperature gradient. Since geostrophic and hydrostatic assumptions are robust, this is also a robust association, one of the most important equations for large scale meteorology 11

Implications for cyclones Cold core lows intensify upward (e.g., extratropical cyclones) Warm core lows weaken upward, and can in fact reverse (e.g. hurricanes) Also for highs 1

Barotropic and baroclinic atmospheres Consider orientation of temperature contours relative to height contours. Barotropic no horizontal (pressure) temperature gradient geostrophic flow independent of height Equivalent barotropic height and thickness contours parallel geostrophic flow changes speed with height Baroclinic temperature variations exist without restriction geostrophic flow can have different speed and direction with height To a good approximation, many aspects of the atmosphere are equivalent barotropic ather than height and temperature, we can think of pressure and density 13

Baroclinic vs. Barotropic Barotropic r=r(p) only ( p T = 0) Baroclinic r=r(p,t) 1) isobaric and isothermal surfaces coincide ) no vertical wind shear (thermal wind = 0) 3) no tilt of pressure systems with height 1) isobaric and isothermal surfaces intersect ) vertical wind shear (thermal wind 0) 3) pressure systems tilt with height Seasons: Geographic: Atmosphere is most baroclinic in winter. Atmosphere is least baroclinic in summer. Atmosphere is most baroclinic in midlatitudes Atmosphere is least baroclinic in the Tropics 1

Study problems What is the divergence of the geostrophic flow? Holton 3.18-3.3 eading ahead, start on chapter.1-. 15