Chapter 3. Fundamentals of Atmospheric Modeling

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1 Overhead Slides for Chapter 3 of Fundamentals of Atmospheric Modeling by Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA January 30, 00

2 Wind Velocity and Speed Velocity Rate at which the position of a body changes with time Velocity vector v = iu+ jv+ kw (3.) Horizontal velocity vector vh = iu+ jv (3.) Scalar components of wind velocity u x = d v y = d w z = d (3.) Wind speed Magnitude of the velocity vector v = u + v + w (3.3) Horizontal wind speed v h = u + v (3.3)

3 Zonally-/Monthly-Averaged Winds (m/s) Figure 3. a January Altitude (km) Latitude

4 Zonally-/Monthly-Averaged Winds (m/s) Figure 3. b July Altitude (km) Latitude

5 Local and Total Differentiation Expansion of total derivative with chain rule d = dx t + x = t +u x (3.4) Total derivative. Time rate of change along a trajectory d Local derivative. Time rate of change at a fixed point t Transport term. Time rate of change due to transport u x Eulerian frame of reference Frame of reference of a fixed point in space t + u x Lagrangian frame of reference Frame of reference of a moving parcel d

6 Example Example 3.. Balloon traveling with the wind from east to west d x = 0 8 molec cm -3 s - = 0 0 molec cm -3 km - u = -0 m s - Find time rate of change in concentration at fixed point A t = d t u d x = 0 8 molec. m molec. km 0 00 = 0 8 molec. 3 cm s s 3. cm km m 3 cm s

7 Gradient Operator Gradient operator in Cartesian / altitude coordinates = i + j + k x y z (3.6) Dot product of velocity vector with gradient operator v = ( iu+ jv+ kw) i + j + k = u + v + w x y z x y z (3.7) i i i = j j=, k k = j= 0 i k = 0 j k = 0 Dot product of gradient operator with velocity vector Divergence term u v v = i + j + k ( iu+ jv+ kw)= + + x y z x y w z (3.8) Dot product of gradient operator with vector not symmetric v v Dot product of two vectors is symmetric a v = v a

8 Gradient Operator Scalar concentration divergence term u ( v)= + v + x y w z (3.9) Gradient of a scalar variable is a vector = + + = + + x y z i j k i j k x y z (3.0) Apply dot product of velocity and gradient operator to ( v ) = u + v + w = u + v + w x y z x y z (3.) Substitute into total derivative d = + ( v ) (3.) t Generalize and expand total derivative d = u v w t = + v (3.3) x y z t

9 Continuity Equation y Figure 3.. u u z x Accumulation = molecules entering - molecules leaving x y z = u y z t u y z t (3.4) Divide both sides by t and box volume ( x y z) t u u = (3.5) x x --> 0 and t --> 0 --> Flux divergence form of continuity equation t u x = (3.6)

10 Continuity Equation Flux divergence form of continuity equation in three dimensions t u v w = = ( v ) (3.7) x y z Take dot product of gradient operator with v ( v)= ( v)+ ( v ) (3.8) Substitute into (3.7) t = ( v) ( v ) (3.9) From definition of total derivative d ( v ) = t (3.) Substitute into (3.9) and similar equation for density --> Velocity divergence forms of continuity equation d = ( v ) (3.) dρa = ρa v (3.3)

11 Continuity Equation for Mass Mixing Ratio umber concentration as a function of moist-air mass mixing ratio = Aρ a q (3.4) m Substitute (3.4) into (3.9) ρ q a t q + ρ ρ ρ a( v)+ ( v ) a + a = ρa( v ) q (3.5) t Substitute continuity equation for air --> Continuity equation for moist-air mass mixing ratio of a species with no external source or sink terms. q t = ( v q ) (3.6)

12 Compressibility / Incompressibility Compressible fluid (air) Volume of an air parcel changes over time and density varies (thus air is inhomogeneous) u v w v = + + = 0 x y z Incompressible fluid (water) Volume of a water parcel stays constant, but density varies (thus water is also inhomogeneous) u x v w + + = 0 (3.7) y z Substitute water density (ρ w ) and v = 0 into (3.3) Density of incompressible fluid is constant along motion dρ w = 0 (3.8) Substitute water density (ρ w ) and v = 0 into (3.0). At a fixed point in the fluid, water density changes. ρw t = v ρw (3.9)

13 Expanded Continuity Equation Expanded continuity equation t D R n n et = + + = v, (3.30) Substitute = = x y z x y z i j k i j k = + + x y z (3.3) into (3.30) to obtain t u x v y w z D x y z R n n et = =, (3.3)

14 Time and Grid Volume Averaging Precise gas concentration = + (3.33) Time and grid volume averaged concentration = t x y z t+ t t x+ x x y+ y y z+ z z d dy x t z d d (3.34) Precise wind velocity vector v = v+ v (3.36) u = u + u v = v + v w = w + w (3.35) Figure 3.3. Precise, mean, and perturbation components of scalar velocity and gas concentration, respectively. u _ u u' _

15 Continuity Equation Substitute (3.33) and (3.35) into (3.3) + t + u u x v v y w w z + + = ( + ) + + D x y Simplify terms + + z + et, R n n= (3.37) + t = (3.38) + t = = + + t t t + = + = = 0 Simplify again + u + u x ( u + u + u + u ) u u x x = + = (3.39) u = 0 u = 0 u = u

16 Continuity Equation Substitute averaged terms into (3.50) t u x v y w z u v w + + x y z = D + + x y et, + R n z n= (3.40) Turbulent diffusion is much larger than molecular diffusion --> ( u ) ( v ) ( w ), u v w = R n t x y z x y z n= (3.4) Continuity equation for gas in vector notation. et t et, + ( v)+ ( v )= R n n= (3.4) Similar equation for air mass density ρa t + ( vρa)+ ( v ρ a)= 0 (3.43)

17 Continuity Equation for Mass Mixing Ratio Continuity equation for trace gas in mass mixing ratio units q t + v q = et, R n n= (3.44) Continuity equation for air ρa t = ρa( v) ( v ) ρa (3.0) Sum continuity equations + = ρaq t et, ρavq ρa Rn n= (3.45) Take time and grid volume average of terms in equation --> Species continuity equation q t + ( v ) q + ( ρav q )= ρa et, Rn n= (3.49)

18 Parameterization of Diffusion Eddies Swirling motions of air caused by wind shear and enhanced by buoyancy Turbulence Many eddies of different sizes acting together Turbulent diffusion Subgrid diffusion due to turbulence K-theory Relates turbulent flux of one parameter (e.g. concentration) to gradient of mean value of the parameter Kinematic turbulent flux terms u = Khxx, x v = Khyy, y w = Khzz, z (3.50) Substitute (3.50) into (3.4) t u x v = y + + z K hzz, z w z x K x y K hxx, + hyy, y et, Rn n= (3.5)

19 Continuity Equations for Trace Gases and Air Continuity equation for trace gas number conc. (molec. cm -3 ) t + et, + ( v)= Kh Rn n= (3.5) Expanded continuity equation for a trace gas q t + ( vq)= ( K h ) q (3.56) + Remisg + Rdepg + Rwashg + Rchemg + Rnucg + Rc eg + Rdp sg + Rds eg + Rhrg Continuity equation for air ρa t + ( vρa )= 0 (3.55)

20 Continuity Equations for Particles Continuity equation for particle number conc. (partic. cm -3 ) ni t + ( vni)= K h ni (3.58) + Remisn + Rdepn + Rsedn + Rwashn + Rnucn + Rcoagn Volume concentration versus number concentration and volume vqi, = niυ qi, (3.57) Continuity equation for particle volume concentration (cm 3 cm -3 ) vqi, t + ( vvqi, )=( K h ) vqi, (3.59) + Remisv + Rdepv + Rsedv + Rwashv + Rnucv + Rcoagv R R R R R R cev dpsv dsev eqv aqv hrv

21 Continuity Equations for Water Water vapor qv + ( v ) qv = ( ρakh ) qv (3.6) t ρa + RemisV + RdepV + RchemV + RnucV + Rc ev + Rdp sv Liquid water q Li, + ( v ) qli, = ( ρakh ) ql, i (3.6) t ρa + RemisL + RdepL + RsedL + RnucL + RcoagL + Rc e L + Rf m L Ice q Ii, + ( v ) qii, = ( ρakh ) qi, i t ρa + RdepI + RsedI + RnucI + RcoagI + Rf mi + Rdp s I (3.63) Bulk vs. Size-Resolved Process Size resolved treatment of water B qt = qv + ql, i + qi, i i= (3.60) Bulk treatment of water qt = qv + ql + qi

22 Thermodynamic Energy Equation Energy transfer processes Conduction Advection Forced Convection Free Convection Turbulence Radiative transfer Energy sources / sinks Latent heat release / absorption Solar / infrared emissions / absorption First law of thermodynamics dq = cpd, dtv α adpa (.76) Differentiate, substitute αa = ρa, rearrange --> Thermodynamic energy equation in terms of temperature along the motion of an air parcel dtv dq = + cpd, cpd, ρa dpa (3.64)

23 Thermodynamic Energy Equation κ Differentiate θv Tv 000 pa with respect to time = dθv = dtv κ κ dp θ d Tv κ a v Tv pa p = a p a Tv (3.65) κθv pa dpa Substitute into (3.64) and expand total derivative --> Thermodynamic energy equation dθv θ θ dq = v + ( v ) θ v v = t cpd, Tv (3.66) Multiply (3.66) by c pd, ρ a, continuity equation by c pd, θ v and sum ( cpd, ρaθv) c ρ θ ρ θ + ( v pd, a v)= a v t Tv dq (3.67) Define energy density (J m -3 ) E = c pd, ρ a θ v Substitute into (3.67) E t + ( ve)= ρa θ v Tv dq (3.68)

24 Thermodynamic Energy Equation Decompose terms (assume ρ a << ρa) ρa ρa θv = θv + θ v v = v+ v Substitute into (3.67) + + ρθ a v ρ θ ρ θ ρ θ a v dq av v ( av v)= t cpd, Tv (3.70) Expand and substitute continuity equation θv θ θ ρ θ v dq + ( v ) v + ( av v)= t ρa cpd, Tv (3.7) Subgrid scale turbulent flux divergence terms θ u v = K v θ θ h, xx v θ = K v v h, yy w θ v = Kh, zz x y (3.73) θv z

25 Thermodynamic Energy Equation Substitute v θ v = Kh θv into (3.7), remove overbars θv θ θ ρ θ v dq + ( v ) v = ( akh ) v + t ρa cpd, Tv (3.74) Rewrite diabatic energy source/sink term eh, θv θ θv ρa h θ v dq + ( v ) = n ( K ) v + t ρa cpd, Tv n= (3.76) where eh, dq dqn dqce dq f m dqdp s dq dq = solar ir = n= (3.75)

Chapter 5. Fundamentals of Atmospheric Modeling

Chapter 5. Fundamentals of Atmospheric Modeling Overhead Slides for Chapter 5 of Fundamentals of Atmospheric Modeling by Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 January 30, 2002 Altitude