ESCI 342 Atmospheric Dynamics I Lesson 10 Vertical Motion, Pressure Coordinates

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1 Reading: Martin, Section 4.1 PRESSURE COORDINATES ESCI 342 Atmosheric Dynamics I Lesson 10 Vertical Motion, Pressure Coordinates Pressure is often a convenient vertical coordinate to use in lace of altitude. If the hydrostatic aroximation is used, the relationshi between ressure and altitude is given by the hydrostatic equation, g z In height coordinates the vertical velocity is defined as coordinates the vertical velocity is defined as D Dt. (1) w Dz Dt. In ressure, (2) and is commonly called simly omega. The units of are Pa/s (often microbars er second, b/s, is also used). Since ressure decreases uward, a negative omega means rising motion, while a ositive omega means subsiding motion. w and are related as follows: D u v w Dt t x y z. (3) On the synotic scale we can assume hydrostatic vertical balance, so that (3) becomes D u v gw gw, (4) Dt t x y since the local ressure tendencies and horizontal ressure advection terms are much, much smaller in magnitude than the vertical ressure advection terms. The total derivative in ressure coordinates is D u v Dt t x y.1 (5) The conversion of a height derivative to a ressure derivative is accomlished using the chain rule as follows z z g z. (6) In ressure coordinates, the directions of the unit vectors (, ĵ, and ) are the same as in height coordinates. The x and y axes are still horizontal, and not oriented along the constant ressure surface. 2 The vertical axis is still vertical (erendicular to x and y.) î ˆk 1 If you take the total derivative of ressure you end u with the seeming absurdity that u v. However, the three artial-derivative terms are actually zero since they are t x y taken holding ressure (the vertical coordinate) constant. 2 See The quasi-static equations of motion with ressure as indeendent variable, A. Eliassen, Geof. Publ., 17, 1949

2 The u and v comonents of the wind are the same in both height and ressure coordinates. MOMENTUM EQUATIONS IN PRESSURE COORDINATES In ressure coordinates the horizontal momentum equation is DVH ˆ k f VH (7) Dt The hydrostatic equation in ressure coordinates is. (8) CONTINUITY EQUATION IN PRESSURE COORDINATES The continuity equation is ressure coordinates is derived by writing the conservation of mass, m, for a arcel as follows: Dm D x y z 0. (9) Dt Dt If the atmoshere is in hydrostatic balance, then z g, so (9) becomes D x y 0. (10) Dt Equation (10) exands out as D x y y D x x D y x y D 0 Dt Dt Dt Dt which can also be written as 1 D 1 D 1 D x y 0. (11) x Dt y Dt Dt From the fact that D D D x u; y v; Dt Dt Dt equation (11) becomes u v 0 x y which in the limit as the arcel becomes infinitesimally small is u v 0. (12) x y Equation (12) is the full continuity equation in ressure coordinates. It contains no assumtions about incomressibility. The full continuity equation in ressure coordinates looks very much like the incomressible continuity equation. This is one of the advantages of using ressure coordinates. THERMODYNAMIC ENERGY EQUATION IN PRESSURE COORDINATES The thermodynamic energy equation in ressure coordinates is 2

3 c DT D J, (13) Dt Dt which exanded out, and using the definition of, becomes T T J V T t c c A B C D E In this form, the terms reresent: Term A Local temerature tendency Term B Horizontal thermal advection Term C Vertical thermal advection Term D Adiabatic exansion/comression due to vertical motion Term E Diabatic heating (radiation, latent heat, etc.) 3. (14) Terms C and D can be combined and written as T, c Rd and defining the static-stability arameter,, as, (15) we get the following form of the thermodynamic energy equation in ressure coordinates. T J V T t R c. (16) A B C D In this form of the equation, the vertical advection and adiabatic exansion/comression are combined into one term, Term C. The static stability arameter is a ositive number for a stable atmoshere, and a negative number for an unstable atmoshere. VERTICAL MOTION Vertical motion is very imortant for forming clouds, and also effects the stability of the atmoshere; however, it is not routinely measured. Therefore, it must be inferred from other measured quantities. Kinematic method for calculating vertical motion One method of calculating the vertical motion is the kinematic method, which integrates the continuity equation between the surface and some ressure level above the surface to get s ( ) ( ) V d. (17) s If the surface is flat and the surface ressure tendency is zero, then (s) = 0 and (17) becomes d

4 s ( ) V d. (18) This gives the exected result that integrated convergence gives uward motion and integrated divergence gives downward motion. The kinematic method has some major flaws. It is only the ageostrohic art of the wind field that can be divergent, and this is very small comared to the actual wind. In fact, the ageostrohic wind is of the order of the errors in the wind observations themselves. This means that divergences calculated from the observed winds may have large errors. Though it isn t much use for calculating actual values of vertical motion, the kinematic method is good for illustrating some general oints about divergence and its relation to vertical motion. Since the vertical motion must disaear at the ground, and is also usually quite small at the to of the trooshere, a grah of the vertical motion with height would look something like that shown below Since must disaear at some level, the divergence also disaears at that level. This leads to the conclusion that There is some level in the atmoshere at which there is no horizontal divergence. This level is known as the level of non-divergence, or LND. Observations indicate that the level of non-divergence usually occurs at around 600 mb. However, since 600 mb is not a standard ressure level for reorting, traditionally meteorologists consider 500 mb to be the level of non-divergence. Adiabatic method for calculating vertical motion Another method for calculating vertical motion uses the thermodynamic energy equation, (16), solved for and assuming adiabatic conditions, to get R T d V T. (19) t A drawback to this method is that temerature tendency must also be known. PRESSURE TENDENCY EQUATION An equation for the change in ressure at a fixed oint in the atmoshere can be derived as follows: Differentiate the hydrostatic equation with resect to time to get 4

5 Substituting for t g. (20) t z t from the continuity equation gives g V gh V g w z t z which when integrated from some level z to the to of the atmoshere yields g V dz gw H t t. (21) z z Since /t at the to of the atmoshere is zero, equation (21) for the ressure tendency at level z is g H V dz gw t z This equation can be exanded as g H V dz g V H dz gw t. (22) z z Using the ideal gas law we can show that R T R T T T so (22) becomes H d H d H H 2 g 1 g g H Vdz V H dz V HT dz gw t R T R T. (23) z d z d z A B C D E The hysical interretation of the ressure tendency equation is as follows: Term A reresents the local ressure tendency Term B reresents the vertically integrated divergence above the level of interest. Integrated divergence above the layer leads to lower ressure. Integrated convergence above the layer leads to higher ressure. Term C reresents integrated advection of ressure. If the winds are in geostrohic or gradient balance, this term will be zero. Term D reresents the integrated temerature advection. Advection of warm air lowers the ressure. Term E reresents advection of mass across the layer. Uward vertical velocity leads to increased ressure, as the mass is moved above the level (since it is the mass above the level that determines the ressure in a hydrostatic atmoshere). At the surface of the Earth, if the surface if level, then Term E would be zero. 5

6 EXERCISES 1. Show that the hydrostatic equation in ressure coordinates is. Hint: Start with z g and use the chain rule and the definition of geootential. 2. Show that Hint: Take T. 0 of T c Rd. 3. If horizontal advection and diabatic heating are negligible, then the local temerature tendency from the thermodynamic energy equation is T. t Rd This equation says that if the atmoshere is stable then downward motion will result in an increase in temerature at a fixed level, while if the atmoshere is unstable then downward motion will result in a decrease in temerature at a fixed level. Give a hysical exlanation as to why this occurs. 4. Use the adiabatic method to estimate the 500 mb vertical velocity () for the following situation. The temerature tendency is zero. The temerature at 600 mb is 13C, at 500 mb it is 19C. The wind at 500 mb is from the SW at 20 m/s, and the temerature at 500 mb increases toward the West at 1C/100 km. 5. For a tyical troical cyclone (which is a warm-core circulation in gradient balance), exlain whether each term in the ressure tendency equation contributes to surface develoment (lower ressures at the surface) or to weakening (higher ressures at the surface). Which terms do you think are most imortant for surface develoment? 6