1	Lecture 3: Applications of Basic Equations Pressure Coordinates: Advantage and Disadvantage Momentum Equation è Balanced Flows Thermodynamic & Momentum Eq.s èThermal Wind Balance Continuity Equation è Surface Pressure Tendency Trajectories and Streamlines Ageostrophic Motion
2	Pressure as Vertical Coordinate From the hydrostatic equation, it is clear that a single valued monotonic relationship exists between pressure and height in each vertical column of the atmosphere. Thus we may use pressure as the independent vertical coordinate. Horizontal partial derivatives must be evaluated holding p constant. è How to treat the horizontal pressure gradient force?
3	Horizontal Derivatives on Pressure Coordinate
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5	Horizontal Momentum Eq. Scaled for Midlatitude Synoptic-Scale
6	Advantage of Using P-Coordinate Thus in the isobaric coordinate system the horizontal pressure gradient force is measured by the gradient of geopotential at constant pressure. Density no longer appears explicitly in the pressure gradient force; this is a distinct advantage of the isobaric system. Thus, a given geopotential gradient implies the same geostrophic wind at any height, whereas a given horizontal pressure gradient implies different values of the geostrophic wind depending on the density.
7	Geostrophic Approximation, Balance, Wind
8	Vertical Velocity in P-Coordinate Vertical Velocity in the Z-coordinate is w, which is defined as dz/dt: w > 0 for ascending motion w < 0 for descending motion Vertical velocity in the P coordinate is ω (pronounced as “omega”), which is defined as dp/dt: ω < 0 for ascending motion ω > 0 for descending motion
9	Continuity Eq. on P-Coordinate Following a control volume (δV= δxδyδz = -δxδyδp/ρg using hydrostatic balance), the mass of the volume does not change:
10	Velocity Divergence Form (Lagragian View)
11	Thermodynamic Eq. on P-Coordinate This form is similar to that on the Z-coordinate, except that there is a strong height dependence of the stability measure (S_p), which is a minor disadvantage of isobaric coordinates.
12	Scaling of the Thermodynamic Eq.
13	Balanced Flows
14	Rossby Number Rossby number is a non-dimensional measure of the magnitude of the acceleration compared to the Coriolis force: The smaller the Rossby number, the better the geostrophic balance can be used.
15	Geostrophic Motion “Geo” è Earth “Strophe” è Turing
16	Natural Coordinate At any point on a horizontal surface, we can define a pair of a system of natural coordinates (t, n), where t is the length directed downstream along the local streamline, and n is distance directed normal to the streamline and toward the left.
17	Coriolis and Pressure Gradient Force Because the Coriolis force always acts normal to the direction of motion, its natural coordinate form is simply in the following form:
18	Acceleration Term in Natural Coordinate
19	Horizontal Momentum Eq. (on Natural Coordinate)
20	Geostrophic Balance
21	Cyclostrophic Balance If the horizontal scale of a disturbance is small enough, the Coriolis force may be neglected compared to the pressure gradient force and the centrifugal force. The force balance normal to the direction of flow becomes in cyclostrophic balance. An example of cyclostrophic scale motion is tornado. A cyclostrophic motion can be either clockwise or counter-clockwise.
22	Gradient Balance Horizontal frictionless flow that is parallel to the height contours so that the tangential acceleration vanishes (DV/Dt = 0) is called gradient flow. Gradient flow is a three-way balance among the Coriolis force, the centrifugal force, and the horizontal pressure gradient force. The gradient wind is often a better approximation to the actual wind than the geostrophic wind.
23	Super- and Sub-Geostrophic Wind
24	Upper Tropospheric Flow Pattern
25	Convergence/Divergence and Vertical Motion
26	Convergence/Divergence in Jetstreak
27	Combined Curvature and Jetstreak Effects
28	Developments of Low- and High-Pressure Centers
29	Trajectory and Streamline It is important to distinguish clearly between streamlines, which give a “snapshot” of the velocity field at any instant, and trajectories, which trace the motion of individual fluid parcels over a finite time interval. The geopotential height contour on synoptic weather maps are streamlines not trajectories. In the gradient balance, the curvature (R) is supposed to be the estimated from the trajectory, but we estimate from the streamlines from the weather maps.
30	Thermal Wind Balance
31	Physical Meanings The thermal wind is a vertical shear in the geostrophic wind caused by a horizontal temperature gradient. Its name is a misnomer, because the thermal wind is not actually a wind, but rather a wind gradient. The vertical shear (including direction and speed) of geostrophic wind is related to the horizontal variation of temperature. è The thermal wind equation is an extremely useful diagnostic tool, which is often used to check analyses of the observed wind and temperature fields for consistency. It can also be used to estimate the mean horizontal temperature advection in a layer. Thermal wind blows parallel to the isotherms with the warm air to the right facing downstream in the Northern Hemisphere.
32	Vertical Motions For synoptic-scale motions, the vertical velocity component is typically of the order of a few centimeters per second. Routine meteorological soundings, however, only give the wind speed to an accuracy of about a meter per second. Thus, in general the vertical velocity is not measured directly but must be inferred from the fields that are measured directly. Two commonly used methods for inferring the vertical motion field are (1) the kinematic method, based on the equation of continuity, and (2) the adiabatic method, based on the thermodynamic energy equation.
33	The Kinematic Method We can integrate the continuity equation in the vertical to get the vertical velocity.
34	The Adiabatic Method The adiabatic method is not so sensitive to errors in the measured horizontal velocities, is based on the thermodynamic energy equation.
35	Barotropic and Baroclinic Atmosphere Barotropic Atmosphere èno temperature gradient on pressure surfaces isobaric surfaces are also the isothermal surfaces density is only function of pressure ρ=ρ(p) no thermal wind no vertical shear for geostrophic winds geostrophic winds are independent of height you can use a one-layer model to represent the barotropic atmosphere
36	Barotropic and Baroclinic Atmosphere Baroclinic Atmosphere ètemperature gradient exists on pressure surfaces density is function of both pressure and temperature ρ=ρ(p, T) thermal wind exists geostrophic winds change with height you need a multiple-layer model to represent the baroclinic atmosphere