Conservation of Energy Thermodynamic Energy Equation

Transcription

1 Conseration of Energy Thermodynamic Energy Equation The reious two sections dealt with conseration of momentum (equations of motion) and the conseration of mass (continuity equation). This section addresses the conseration of energy. The first law of thermodynamics, of which you should be ery familiar with from thermodynamics class, is the conseration of energy law. The most common form of the 1 st law in meteorology is the enthaly form, which is written as: = C dt αd (1) where = change in heating, C dt = change in internal energy (where C is the secific heat at constant ressure), and αd = change in energy due to the work of exansion. Adding αd to both sides of (1) and then diiding by C yields: dt d 1 = α + (2) C C (a) (b) (c) with the secified terms defined as: a) rate of change of temerature (T) with time inside an air arcel b) rate of change of temerature with time due to work of exansion c) rate of change of T with time due to diabatic heating This is nice, but we robably want a more useful form if we are trying to redict temerature in a numerical model. Now we know from Euler s relation that the total deriatie (a) can be broken down into the local rate of change and adection comonents. After the substitution and some rearranging, Eq. (2) becomes: α d 1 = u w + + t x y z C C (a) (b) (c) (d) (e) (f) (3) a) rate of change of T at a grid oint Remember, this is the Eulerian form now b) zonal adection of T The sign of this term deends on the sign of the temerature gradient. Is warmer or cooler air being moed in? c) meridional adection of T Usually, temerature decreases as latitude increases ( < 0 ). So if there is a south wind ( > 0), this term will contribute ositiely to the y local temerature change. Warm air is being adected in.

2 d) ertical transfer of T In most cases the temerature decreases with height in the atmoshere ( < 0 ). Uward elocity (w > 0) will adect warmer air. z e) adiabatic temerature change f) diabatic temerature change Thermodynamic Energy Equation If you watch the Weather Channel, you will frequently hear the meteorologist say There is a lot of energy associated with this system, or Most of the energy is concentrated to the north of the frontal boundary. Are they referring to the winds? Probably not. Usually they are alluding to an area with intense reciitation and/or conection an area with a lot of latent heat release. Heating and kinetic energy are intimately linked. For examle, if an air arcel is warmed, it becomes buoyant and begins to rise, acquiring kinetic energy. Also, heating affects the kinetic energy of molecules at a microscoic leel. Molecules will ibrate more igorously as they are heated. We hae a term called thermodynamic energy which wras u this internal energy (ibrating molecules) with the kinetic energy (wind). It is defined as the sum of these two comonents. The equation which describes this relationshi, called the thermodynamic energy equation, is deried beginning with an alternatie form of the 1 st Law of Thermodynamics, the internal energy form: = due + dα (4) After some rearranging, diiding by, and substituting α = 1/ we get: 1 du d e = + (5) Mathematics tells us that 1 d 1 = 2 d. Equation (5) becomes: Now multily both sides by : du e d = + (6) 2

3 du e d = + (7) Now things get really silly. Do you see anything in the first term on the right side that can be substituted for? Hint? Well, here s the continuity equation that you hae already forgotten about: d u w α = + + or, x y z 1 d = V Now do you see anything we can substitute for? Then do it! Eq. (7) becomes: du e = V + (8) This equation is called the thermal energy equation, and it is worth a coule minutes to oint out some things. It tells us how the internal energy changes with time (er unit olume). The change of internal energy deends uon the rate at which work is done ( V term) on the olume and the rate at which energy is added ( ). The work done on the olume is related to the elocity diergence. If there is a net diergence ( V > 0 ), equation 8 tells us that the change in internal energy < 0. In other words, diergence leads to exansion of the olume (which requires work), lowering the internal energy. The oosite can be said of elocity conergence, which will warm the arcel. The thermal energy equation makes u half of the thermodynamic energy equation. Now we need to find a relationshi for the change in mechanical (kinetic energy) with heating. Where better to start than Newton s second law of motion in ector form! dv = α 2Ω V + g + (9) Multily both sides of (9) by and then dot multily both sides by V to yield: dv V = V ( ) + V ( 2 Ω V ) + V g + V Fr (10) F r

4 Now we can simlify a bit. The rotation term ( V ( 2Ω V ) ) becomes = 0 because the cross roduct creates a third ector that is erendicular to the original two ectors. Dot multilying the wind ector with the erendicular ector = 0. I know, you are uttering exlicaties right now but go back to the math section if you need to reiew. The term V g can be written as: + (11) ( ui j + wk ) ( gk ) since graity only acts in the k direction (and is negatie because it acts in the negatie k direction). When you carry out the dot roduct, remember that a unit ector that is dotted with itself = 1 and one dotted with any other unit ector = 0. Thus, (11) becomes: ( ui + j + wk ) ( gk ) = gw Now things get really fun. We know that the ertical elocity (w) = dz/. Also, try to recall something else that you robably already forgot, which is that gdz = dφ (change in geootential). So, (12) dz dφ gw = g = (13) This term is now essentially the graitational otential energy. Energy is needed to lift arcels into the atmoshere, and this term accounts for that. Substituting (13) into (10), we get: dv dφ V = V ( ) + V Fr (14) We are almost home. The term on the LHS of (14) is the kinetic energy and can be rewritten as: 1 d V V dv V = 2 (15) Sustitute (15) into (14) and rearrange a little to get: 1 d V V dφ 2 + = V ( ) + V ( Fr ) (16) Finally, combine the two terms on the LHS to yield:

5 1 d V V + Φ 2 = V (a) (b) (c) ( ) + V ( ) F r (17) This is the mechanical energy equation. In English, it says that the sum of the kinetic and geootential energy change with time is equal to the roduction of energy by the PGF (b) and the dissiation of energy from the friction force (c). Note that the units for all terms in this equation are the same as the thermal energy equation. Now we want to combine the thermal energy equation (8) with the mechanical energy equation (17) to create the thermodynamic energy equation. Recall the thermal energy equation: du e = V + (8) Add this to equation (17) to yield: 1 d V V du + Φ e 2 + = V + V ( ) + V ( Fr ) + (18) You can entertain yourself by showing that: V + V ( ) = ( V ) Substituting, rearranging, and moing the graitational otential energy back to the RHS gies us: du e 1 d V V 2 dφ + = ( V ) + V ( Fr ) + Now we can combine the two terms on the LHS, multily through by olume ( δ V ), and dφ reert the - to V g gies us a form of the thermodynamic energy equation: d u e 1 + V V δv = V gδv VδV + V r 2 t (a) (b) (c) (d) (e) q ( F ) δv δv (19)

6 a) rate of change of the internal energy and kinetic energy b) graitational acceleration c) ressure gradient force d) frictional deceleration e) diabatic heating So we see that internal and kinetic energy is directly controlled by the not only the forces of graity, friction, and the PGF but also by the amount of heating that occurs within a gien olume.