ATM 298, Spring 2013 Lecture 2 The Equa;ons of Fluid Mo;on April 3, Paul A. Ullrich (HH 251)

Transcription

1 ATM 298, Spring 2013 Lecture 2 The Equa;ons of Fluid Mo;on April 3, 2013 Paul A. Ullrich (HH 251) paullrich@ucdavis.edu

2 Global Atmospheric Modeling Global atmospheric models were originally constructed as a means of understanding the general circulation of the atmosphere. They have been used as predictive models on timescales of days to weeks (for numerical weather prediction) up to centuries (long-term climate forecasts). Atmospheric models are also a tool which allow for experiments to be performed on the Earth system. They have been used as laboratories for studying paleoclimate, planetary atmospheres (Mars, Titan, Jupiter), and answering scientific questions regarding the drivers of weather and climate.

3 Anatomy of an Atmospheric Model Dynamical Core We will start here Physical ParameterizaFons

4 The Equa;ons of Mo;on

5 Atmospheric Dynamics The principles of atmospheric dynamics are largely drawn from fluid dynamics (mathematically based). The physics are generally simple: Conservation of momentum Conservation of mass Conservation of energy Newtonian physics: Newton s laws of motion applied to the atmosphere: Force = Mass x Acceleration Acceleration = Change of velocity with time Velocity = Change of position with time

6 Atmospheric Dynamics Newtonian physics: Newton s laws of motion applied to the atmosphere: Force = Mass x Acceleration Acceleration = Change of velocity with time Velocity = Change of position with time F = ma a = dv v = dx This is the starfng point.

7 Conservation There are certain parameters (energy, momentum, mass, air, water, ozone, number of atoms, etc.) that must be conserved. Conservation means that in an isolated system that parameter remains constant. It s not created. It s not destroyed. But it can move around. Is the Earth s atmosphere isolated? The Earth s atmosphere is not exactly isolated (escape to space, escape beneath the surface), but to a close approximation this is the case!

8 Conservation For example, if we don t spend money or make money then the money we have tomorrow is the same as the money we had yesterday. M tomorrow = M yesterday This is not a particularly realistic model, of course But it is the essence of a conservation or continuity principle

9 Conservation Conservation (continuity) principle with production and loss terms Monthly Income My Bank Account $ Monthly Expense M tomorrow = M yesterday + N(I E) N= number of months

10 Conservation Conservation (continuity) principle with production and loss terms M tomorrow = M yesterday + N(I E) Algebra: (M tomorrow M yesterday )=N(I E) How the difference changes with time: M tomorrow N M yesterday = I E Change of money with Fme ProducFon / Loss rates

11 Conservation Conservation (continuity) principle with production and loss terms: dm = Production Loss

12 Basics of the Earth s Atmosphere Atmosphere is composed of air, which is a mixture of gases (treated as an ideal gas). Below an altitude of 100 km, the atmosphere behaves as a fluid (under the continuum hypothesis). p = R d T Ideal gas law p R d T Pressure (Pa) Density (kg/m 3 ) Ideal gas constant for dry air (287 J/kg/K) Temperature (Kelvin)

13 Radius of the Earth km Atmosphere Depth 100 km Troposphere Depth 10 km Mountain Height 5 km

14 Spherical Coordinates Spherical coordinates: Longitude j k r LaFtude Radius Basis vectors: j i Eastward basis vector i j k Northward basis vector VerFcal basis vector

15 Atmospheric Dynamics Newton s Second Law of Motion: a = F m du = F m u =(u, v, w) u v w 3D velocity vector Eastward velocity (zonal velocity) Northward velocity (meridional velocity) Upward velocity (verfcal velocity)

16 Atmospheric Dynamics What are the forces? a = F m du = F m Pressure gradient force Gravitational force Viscous force Coriolis and centrifugal force Total force is the sum of all these forces

17 Atmospheric Dynamics How are these forces expressed? We assume the existence of an idealized parcel of fluid. Forces are calculated on the idealized parcel. Take the limit of the parcel being infinitely small.

18 Pressure Gradient Force z Δz Density (kg/m 3 ) V = x y z Volume (m 3 ) y Δx Δy m = x y z Mass (kg) x p Pressure (Pa) = Force per unit area

19 Pressure Gradient Force z (x 0,y 0,z 0 ) Δz Pressure at parcel center: p 0 = p(x 0,y 0,z 0 ) Approximate pressure here via Taylor expansion p = p x 2 + O( x2 ) y Δx Δy x

20 Pressure Gradient Force p L x 2 Δz p R x 2 Δx Δy F L = p L A L = p L y z F R = p R A R = p R y z F tot = F L + F R = x 2 y z apple p x 2 x y z Total force acfng on fluid parcel in the x direcfon y z

21 Pressure Gradient Force Repeat in all coordinate direcfons: F k x y z = (rp) x y z Then compufng the force per unit mass (recall this determines the accelerafon): F tot m = 1 rp Total force acfng on fluid parcel in all direcfons

22 Gravitational Force Recall Newton s law of gravity: F g = G Mm r 2 r r r But since the atmosphere is essenfally a thin shell, we can make the approximafon r a Define gravity at surface: g = GM a 2 F g m = gk Total gravitafonal force acfng on fluid parcel

23 Coriolis / Centrifugal Force The Earth revolves around its axis at a certain rate. j k Coriolis and Centrifugal forces are known as apparent forces, because they only exist because the reference frame is in mofon. The Coriolis force deflects fluid parcels as a consequence of the Earth s rotafon. The Centrifugal force abempts to push fluid parcels away from the axis of rotafon.

24 Coriolis Force

25 Coriolis Force Sadly, class Fme does not allow for a rigorous derivafon of the mathemafcal form of the Coriolis force. Instead see Holton, for example. j k Basic idea: Coriolis force is proporfonal to rotafon rate and normalized distance from the rotafon axis (Coriolis parameter) f =2 sin DeflecFon is to the right in the Northern hemisphere and to the led in the Southern hemisphere. du = fv dv = fu

26 Coriolis Force DeflecFon is to the right in the Northern hemisphere and to the led in the Southern hemisphere. du = fv dv = fu Tendency is for fluid parcels to move in circles.

27 Centrifugal Force j k Fcent Centrifugal force always works perpendicular to the axis of rotafon: Fcent mv 2 = R r Typically small, so absorbed into the gravitafonal term. Paul Ullrich ATM 298: Lecture 02 April 3, 2013

28 Viscosity The viscosity of air is responsible for resisfng mofon of the fluid. It is a dissipafve force, which results slowing a fluid which is not otherwise forced. F visc m = r2 u Note Laplacian (second derivafve). Why is this the case? Away from the surface viscosity is typically small, and so is ignored.

29 Dynamic Equations of Motion du uv tan r dv + u2 tan r dw + uw r = 1 r cos + vw r = u 2 + v 2 = +2 v sin 2 w cos + r2 u 2 u sin g +2 u cos + r 2 v + r 2 w Curvature Pressure Gradient Gravity Coriolis Viscosity Generally very small

30 Lagrangian / Eulerian Frame Everything we have derived so far has been situated in the Lagrangian frame. Namely, our equafons of mofon are valid following a fluid parcel. This is the natural domain for the material derivafve: d

31 Lagrangian / Eulerian Frame But atmospheric models are built in the Eulerian frame. That is, we are sihng at a parfcular point and are watching the local state of the fluid. In pracfce, tracking individual fluid parcels is very difficult, because deformafon can be very extreme:

32 Material Derivative Consider a background wind field superimposed on an arbitrary temperature field (note that this analysis is valid for quanffes other than temperature as well).

33 Material Derivative Consider a fluid parcel transported passively by this background flow. Δx Δy Expand the change in temperature following the fluid parcel using a Taylor series expansion. Change in temperature following a fluid parcel: z+ Higher Order Terms Eulerian Frame

34 Material Derivative Divide through by t Δy T @x x y z t Δx Take the limit for small @x t dz Note: dx is the rate of change in the x- coordinate following the flow dx = u

35 Material Derivative Final form: Δy + Δx Or in vector notafon: + u rt Lagrangian Frame (following fluid parcel) Eulerian Frame (at a fixed point)

36 Material Derivative Or in vector notafon: + u rt u rt Advec;on of temperature at a point This is the same as taking the derivafve of T along the line defined by the wind vector (and mulfplying by - 1). Warm air advecfon: u rt <0 (temperature increases in the direcfon the wind is blowing from)

37 Dynamic Equations of Motion du uv tan r dv + u2 tan r dw + uw r = 1 r cos + vw r = u 2 + v 2 = +2 v sin 2 w cos + r2 u 2 u sin g +2 u cos + r 2 v + r 2 w Material derivafve: + u r

38 Continuity Equation The confnuity equafon simply expresses conservafon of mass: dm =0 (mass within a fluid parcel does not change) But mass is not a state variable. Instead use density: = m V Then by chain rule: d = 1 V dm m dv V 2 = V dv What is this?

39 Continuity Equation d = 1 V dm m dv V 2 = V dv Change in density due to change in volume of the fluid parcel What causes a fluid parcel to change in volume? Compression (convergence of the velocity field) Expansion (divergence of the velocity field)

40 Continuity Equation d = 1 V dm m dv V 2 = V dv Change in density due to change in volume of the fluid parcel The change in volume is given by the divergence of the field: 1 V dv = r u Hence we obtain the confnuity equafon: r u + r ( u) =0 (Lagrangian form) (Eulerian form)

41 Tracer Transport Passive tracers (chemical species, moisture) are transported with the flow. Their mixing rafo (the rafo of tracer mass to fluid mass) is constant within a fluid parcel: dq i =0 (AdvecFon equafon) To write a conservafon law for tracers, we use the + u r = q i + q i u r = q i r + u rq ( q i)+r ( q i u)=0 (Flux- Form)

42 Thermodynamic Equation Recall the first law of thermodynamics: du W = Q With internal energy U, heafng Q and work W. du = c v dt W = pd Q = J Where = 1 is the specific volume (volume / mass). Finally, divide by : c v dt + pd = J HeaFng / Cooling Change in temperature of fluid parcel Work done on fluid parcel

43 du The Primitive Equations uv tan r dv + u2 tan r dw + uw r = 1 r cos + vw r = u 2 + v 2 = d = c v dt + pd = J r u p = R d T dq i = +2 v sin 2 w cos + r2 u 2 u sin g +2 u cos + r 2 v + r 2 w Material derivafve: + u r

44 Approxima;ons

45 Approximations The complete set of equafons of mofon are typically too complicated to solve in pracfce, so approximafons are made to simplify the mathemafcs. The simplest approximafons include the shallow water equa<ons (one layer incompressible fluid used for ocean simulafons) and Boussinesq fluid (approximate the density to constant almost everywhere). du uv tan r dv + u2 tan r dw + uw r = 1 r cos + vw r = u 2 + v 2 = +2 v sin 2 w cos + r2 u 2 u sin g +2 u cos + r 2 v + r 2 w

46 Euler Equations The viscous scale for air is generally on the order of one meter or less, well below the resolufon of atmospheric models. Consequently, viscious dissipafon is irrelevant outside the boundary layer and so is ignored in the equafons of mofon. Ignoring viscosity leads to the Euler equafons. du uv tan r dv + u2 tan r dw + uw r = 1 r cos + vw r = u 2 + v 2 = +2 v sin 2 w cos + r2 u 2 u sin g +2 u cos + r 2 v + r 2 w

47 Radius of the Earth km Atmosphere Depth 100 km Troposphere Depth 10 km Mountain Height 5 km

48 Shallow Atmosphere Approximation The atmosphere is closely approximated as a thin shell with thickness much smaller than the radius of the Earth. Consequently, the shallow atmosphere approximafon is one of the first used in operafonal models. du uv tan ar dv + u2 tan ar dw + uw r = 1 a r cos + vw r = a u 2 + v 2 = +2 v sin 2 w cos + r2 u 2 u sin g +2 u cos + r 2 v + r 2 w

49 Hydrostatic Approximation Let s turn our abenfon to the verfcal momentum equafon. Typical scales (in m/s 2 ): dw u 2 + v 2 r g +2 u cos + r 2 w

50 Hydrostatic Approximation Typical scales (in m/s 2 ): dw u 2 + v 2 r g +2 u cos + r 2 w Accurate computafon of the verfcal accelerafon requires nearly exact cancellafon of larger terms. This equafon is generally very bad for accurately approximafng the verfcal velocity.

51 Hydrostatic Approximation To a good approximafon, the verfcal velocity equafon typically used as a constraint equafon. du uv tan r dv + u2 tan r dw + uw r = 1 r cos + vw r = u 2 + v 2 = +2 v sin 2 w cos + r2 u 2 u sin g +2 u cos + r 2 v + r 2 w VerFcal velocity is instead diagnosed from the other equafons in the system. This approximafon is fairly accurate as long as the grid spacing is > 10 km.

52 We did it