Balanced flow
Things we know Primitive equations are very comprehensive, but there may be a number of vast simplifications that may be relevant (e.g., geostrophic balance). Seems that there are things in the atmosphere which spin (like highs and lows, and jets around the poles,. other things), and we d like to be able to explain them (quantitatively) Some observations
Dust devils, water spouts, tornados
What could be better than 1 tornado?
Mars Image from Mars Reconnaissance Orbiter Image from Sprit rover
Some puzzles/questions Is geostrophic balance appropriate for all atmospheric motions of interest? Why are high pressure systems usually larger than lows?
Isobaric coordinates From scaling vertical momentum equation, found hydrostatic balance to be very robust Thus define coordinate system x, y, p (recall dx = a cos d and dy = a d) Vertical velocity omega =dp/ (note < 0 means up as dp ~ -dz) Thus expand total (Lagrangian) derivative: d t u x v y p 11
Momentum equation Prom previous lecture, we showed: (more generally, (1/)p = ) Thus, (scaled) momentum equation: 1 dv dp dx z d dx fk V p Density now not present, as coordinate surfaces describe vertical distribution of mass One consequence is horizontal gradients of geopotential at different altitudes have the same geostrophic wind speed (cf. wind speeds is pressure gradient divided by density in height coordinates) 12
Thermodynamic equation Making use of definition of omega: Defining a stability parameter, S: c dt S p dt S J c p T p J d g Notice the stability increases rapidly with height as density decreases States that if the atmosphere is very stable, more work is done for the same parcel expansion Heating directly related to vertical motion 13
Continuity equation Consider a Lagrangian parcel (following the motion) The mass remains constant, but the volume can change. i.e., dp = -gdz. So dv=-dxdydp/(g) and dm=dxdydp/g d( dm) 1 dx d( dx) d( dv ) 1 d( dxdydp) 0 g 1 d( dv) 1 d( dp) 0 dy dp u left u l u r u right u x v y V 0 p 0 dz,u dx dy Thus the remarkably simple result. No density! (compare with x, y, z derivation in from earlier lecture) 14
Divergence (isobaric) u x v y 0 p 15
16 Primitive equations in isobaric coordinates c p J p T p T y T v x T u t T 0 p y v x u p RT friction x fv p u y u v x u u t u friction y fu p v y v v x v u t v g p Notice, additional terms if spherical geometry is used If we consider a moist atmosphere T is the virtual temperature. These equations have been known for over a century System has no known analytic solution Numerical solution is the basis of weather and climate models
Primitive equations are lovely, but They are big and nasty, and we would prefer something more convenient Also, there may be some motions which can by explained more simply
Some puzzles/questions Is geostrophic balance appropriate for all atmospheric motions of interest? Why are high pressure systems usually larger than lows?
Recall: Geostrophic balance Coriolis and pressure gradient are about the same size 1 p x fv 1 p y g fu g Very simple version of the description of momentum However, remarkably (90%) accurate Suggests that flow (u and v) is aligned perpendicular to the pressure gradient force i.e., flow along pressure contours on height surfaces, or flow along height contours on pressure surfaces
Horizontal scaling Acceleration ~ 10% size of Coriolis and PGF, removing it keep about 90% of the description du dv uv tan a u 2 tan a uw a vw a 1 p x 1 p y fv 2wcos fu F ry F rx
Scales Some assumptions for our flows: Horizontal (i.e., no vertical velocity). Steady (i.e., speed does not change) Hurricanes and Tornadoes/dust devils
Natural (intrinsic) coordinates Simplify momentum equations by considering forces in: 1. direction (tangent) of flow, t 2. normal to flow, n 3. vertical, k (but we focus on horizontal) So, V = Vt V = ds/ This is a Lagrangian coordinate, in that it changes following the motion 22
Natural coordinates Consider change in velocity, and change in coordinate system (as we did for Earth s rotating coordinate) dv dvt t dv V Geometry shows that, V n R dv t dv V n R 2 change in speed centripetal acceleration c.f. dv i du j dv R is the radius of curvature, and defined positive to the left (i.e., a right-handed coordinate) 23
Force balance Pressure gradient force: Coriolis force (normal to flow): -fvn t n s n Thus tangent and normal components of momentum equation 2 dv V fv s R n Change in speed due to pressure gradient in direction of flow Thus balanced flow (dv/ = 0) must be at constant height (flow follows contours of geopotential height) If latitudinal variations in f can be neglected, constant geopotential gradient normal to flow implies constant R (flow is circular) 24
Gradient wind balance V R 2 fv n Balanced flow (no friction) Balance = constant speed No change in pressure following trajectory. So all isobaric. Solving for wind speed: V fr 2 f R 4 2 2 R n V defined as positive, and must be real, thus four possible flow configurations 25