The simplifed momentum equations

1. The simplifed momentum equations Height coordinates Pressure coordinates

2. In this section we will derive the four most important relationships describing atmospheric structure • Hydrostatic Balance: the vertical force balance in the atmosphere • The Hypsometric Equation: the relationship between the virtual temperature of a layer and the layer’s thickness • Geostrophic Balance: the most fundamental horizontal force balance in the atmosphere • 4. Thermal wind Balance: a relationship between the wind at an upper level of the atmosphere and the temperature gradient below that level

3. We will then expand our understanding of balanced flow to include more complicated force balances in horizontal flows • Geostrophic Flow: the flow resulting from a balance between the PGF and Coriolis force in straight flow • Inertial Flow: curved flow resulting when the Coriolis force balances the “Centrifugal” force • Cyclostrophic Flow: Flow that results when the pressure gradient force balances the Centrifugal force • 4. Gradient flow: Flow that results when the pressure gradient force and Coriolis force balance the Centrifugal force

4. Virtual Temperature: The temperature that a parcel of dry air would have if it were at the same pressure and had the same density as moist air. • P = pressure • rd = dry air density • rv = vapor density • = air density R= gas constant • Rv = vapor gas constant • Rd = dry air gas constant • T = Temperature Derivation: Start with ideal gas law for moist air: Now treat moist air as if it were dry by introducing the virtual temperature Tv What is the relationship between the temperature, T And the virtual temperature Tv?

5. M = mass of air Md = mass of dry air Mv = mass of vapor V = volume Write: Multiply and divide second term by Rd Cancel Rd and rearrange

6. From last page: Cancel V, and divide top and bottom terms by Md: Introduce mixing ratio: rv = Mv/Md and let e = Rd/Rv

7. From last page: Approximate (1+rv)-1 = 1- rv and use 1/e = 1.61 Neglect term

8. HYDROSTATIC BALANCE The atmosphere is in hydrostatic balance essentially everywhere except in core regions of significant storms such as hurricanes and thunderstorms

9. The Hypsometric Equation Consider a column of atmosphere that is 1 m by 1 m in area and extends from sea level to space Let’s isolate the part of this column that extends between the 1000 hPa surface and the 500 hPa surface How much mass is in the column? How thick is the column? That depends…………

10. Hydrostatic equation Ideal Gas Law Substitute ideal gas law into hydrostatic equation Integrate this equation between two levels (p2, z2) and (p1, z1)

11. From previous page Problem: Tv varies with altitude. To perform the integral on the right we have to consider the pressure weighted column average virtual temperature given by: We can then integrate to give This equation is called the Hypsometric Equation The equation relates the thickness of a layer of air between two pressure levels to the average virtual temperature of the layer

12. Geopotential Height We can express the hypsometric (and hydrostatic) equation in terms of a quantity called the geopotential height Geopotential (): Work (energy) required to raise a unit mass a distance dz above sea level Meteorologists often refer to “geopotential height” because this quantity is directly associated with energy to vertically displace air Geopotential Height (Z): g0 is the globally averaged Value of gravity at sea level For practical purposes, Z and z are about the same in the troposphere

13. Atmospheric pressure varies exponentially with altitude, but very slowly on a horizontal plane – as a result, a map of surface (station) pressure looks like a map of altitude above sea level. observed surface pressure over central North America variation of pressure with altitude

14. Station pressures are measured at locations worldwide Analysis of horizontal pressure fields, which are responsible for the earth’s winds and are critical to analysis of weather systems, requires that station pressures be converted to a common level, which, by convention, is mean sea level.

15. BUT WHAT IS Tv? WE HAVE ASSUMED A FICTICIOUS ATMOSPHERE THAT IS BELOW GROUND!!! Reduction of station pressure to sea level pressure: Integrate hypsometric equation sea level pressure (PSL) to station pressure (PSTN) and right side from z = 0 to station altitude (ZSTN). (4) (5)

16. National Weather Service Procedure to estimate Tv • Assume Tv = T • 2. Assume a mean surface temperature = average of • current temperature and temperature 12 hours earlier • to eliminate diurnal effects. • 3. Assume temperature increases between the station and sea level of • 6.5oC/km to determine TSL. • 4. Determine average T and then PSL. In practice, PSL is determined using a table of “R” values, where R is the ratio of station pressure to sea level pressure, and the table contains station pressures and average temperatures. Table contains a “plateau correction” to try to compensate for variations in annual mean sea level pressures calculated for nearby stations.

17. Implications of hypsometric equation Consider the 1000—500 hPa thickness field. Using Ballpark number: A decrease in average 1000-500 hPa layer temperature of 1K leads to a reduction in thickness of the layer of 60 hPa

18. Implications of Hypsometric Equation A cold core weather system (one which has lower temperature at its center) will winds that increase with altitude

19. How steep is the slope of The 850 mb pressure surface in the middle latitudes? 1200 meters @ 80oN 1630 meters @ 40oN 40o 60 nautical miles/deg  1.85 km/nautical mile = 4452 km Slope = 430 m/4452000 m ~ 1/10,000 ~ 1 meter/10 km

20. 9,720 meters @ 80oN How steep is the slope of The 250 mb pressure surface in the middle latitudes? 10,720 meters @ 40oN 40o 60 nautical miles/deg  1.85 km/nautical mile = 4452 km Slope = 1000 m/4452000 m ~ 1/4,000 ~ 1 meter/4 km

21. GEOSTROPHIC BALANCE A state of balance between the pressure gradient force and the Coriolis force Geostrophic balance Air is in geostrophic balance if and only if air is not accelerating (speeding up, slowing down, or changing direction). For geostrophic balance to exist, isobars (or height lines on a constant pressure chart) must be straight, and their spacing cannot vary.

22. Geostrophic wind The wind that would exist if air was in geostrophic balance The Geostrophic wind is a function of the pressure gradient and latitude In pressure coordinates, the geostrophic relationships are given by

23. Where on this map of the 300 mb surface is the air in geostrophic balance?

24. Geostrophic Balance and the Jetstream Take p derivative: Substitute hydrostatic eqn THE RATE OF CHANGE OF THE GEOSTROPHIC WIND WITH HEIGHT (PRESSURE) WITHIN A LAYER IS PROPORTIONAL TO THE HORIZONTAL TEMPERATURE GRADIENT WITHIN THE LAYER

25. We can write these two equations in vector form as or alternatively

26. The “Thermal Wind” vector The “Thermal Wind” is not a wind! It is a vector that is parallel to the mean isotherms in a layer between two pressure surfaces and its magnitude is proportional to the thermal gradient within the layer. When the thermal wind vector is added to the geostrophic wind vector at a lower pressure level, the result is the geostrophic wind at the higher pressure level

27. A horizontal temperature gradient leads to a greater slope of the pressure surfaces above the temperature gradient. More steeply sloped pressure surfaces imply that a stronger pressure gradient will exist aloft, and therefore a stronger geostrophic wind.

28. Note the position of the front at 850 mb……

29. ….and the jetstream at 300 mb.

30. Implications of geostrophic wind veering (turning clockwise) and backing (turning counterclockwise) with height Warm air Cold air Winds veering with height – warm advection

31. Implications of thermal wind relationship for the jetstream Note the position of the jet over the strongest gradient in potential temperature (dashed lines) (which corresponds to the strongest temperature gradient)

32. Implication of the thermal wind equation for the general circulation Upper level winds must be generally westerly in both hemispheres since it is coldest at the poles and warmest at the equator

33. Natural Coordinates and Balanced Flows To understand some simple properties of flows, let’s consider atmospheric flow that is frictionless and horizontal We will examine this flow in a new coordinate system called “Natural Coordinates” In this coordinate system: the unit vector is everywhere parallel to the flow and positive along the flow the unit vector is everywhere normal to the flow and positive to the left of the flow

34. The velocity vectoris given by: The magnitude of the velocity vector is given by: where s is the measure of distance in the direction. The acceleration vector is given by: where the change of with time is related to the flow curvature.

35. We need to determine The angle is given by Where R is the radius of curvature following parcel motion directed toward center of curve (counterclockwise flow) directed toward outside of curve (clockwise flow)

36. Note here that points in the positive direction in the limit that approaches 0.

37. Let’s now consider the other components of the momentum equation, the pressure gradient force and the Coriolis Force Coriolis Force: Since the Coriolis force is always directed to the right of the motion Since the pressure gradient force has components in both directions PGF: The equation of motion can therefore be written as:

38. Let’s break this up into component equations: To understand the nature of basic flows in the atmosphere we will assume that the speed of the flow is constant and parallel to the height contours so that Under these conditions, the flow is uniquely described by the equation in the yellow box

39. Centrifugal Force PGF Coriolis Force Geostrophic Balance: PGF = COR Inertial Balance: CEN = COR Cyclostrophic Balance CEN = PGF Gradient Balance CEN = PGF + COR

40. Geostrophic flow Geostrophic flow occurs when the PGF = COR, implying that R For geostrophic flow to occur the flow must be straight and parallel to the isobars Pure geostrophic flow is uncommon in the atmosphere

41. Inertial flow Inertial flow occurs in the absence of a PGF, rare in the atmosphere but common in the oceans where wind stress drives currents This type of flow follows circular, anticyclonic paths since R is negative Time to complete a circle: • is one half rotation  is one full rotation/day called a half-pendulum day

42. Power Spectrum of kinetic energy at 30 m in the ocean near Barbados (13N)

43. Cyclostrophic flow Flows where Coriolis force exhibits little influence on motions (e.g. Tornado)

44. Cyclostrophic flows can occur when the Centrifugal force far exceeds the Coriolis force , a number called the Rossby number A synoptic scale wave: NO A tornado: YES Venus (rotates every 243 days): YES

45. In cyclostrophic flow around a low, circulation can rotate clockwise or counterclockwise (cyclonic and anticyclonic tornadoes and smaller vortices are observed) WHAT ABOUT A HIGH???

46. Venus has a form of cyclostrophic flow, but more is going on that we don’t understand

47. Gradient flow This expression has a number of mathematically possible solutions, not all of which conform to reality

48. the unit vector is everywhere normal to the flow and positive to the left of the flow  is the geopotential height is the height gradient in the direction of R is the radius of curvature following parcel motion directed toward center of curve (counterclockwise flow) directed toward outside of curve (clockwise flow)

49. V is always positive in the natural coordinate system Solutions for For radical to be positive Therefore: must be negative. V = negative = UNPHYSICAL

50. Solutions for Positive root physical Radical > Negative root unphysical Anticyclonic Called an “anomalous low” it is rarely observed (technically since f is never 0 in mid-latitudes, anticyclonic tornadoes are actually anomalous lows outward Increasing in direction (low)