Needs work : define boundary conditions and fluxes before, change slides Useful definitions and conservation equations

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1 Needs work : define boundary conditions and fluxes before, change slides Useful definitions and conservation equations

2 Turbulent Kinetic energy The fluxes are crucial to define our boundary conditions, in particular at the surface (earth or ocean) (Mean or advective) Flux of a generic quantity ϕ is the amount of ϕ passing (advected) through a unit area in a unit time (think.e.g. of a mass flux eolian transport)

3 Heat Moisture and Momentum fluxes p11-13bb Note that fluxes are defined in each direction: to define boundary conditions and study the effect of near surface processes on atmospheric turbulence vertical fluxes are of utmost importance Note that vertical mean fluxes vanish in flat terrain (W=0), while turbulent fluxes provide a net contribution 0

4 Let us consider the mixing effect of an eddy under different thermal regimes w θ > 0 w θ < 0

5 Turbulent kinetic energy profiles (thermal stability dependent)

6 a) convective mixed layer: large turbulent fluxes mixing the ABL. w θ > 0 (thermal plumes) maintain the mixed layer b) stably stratified layer; weak turb. fluxes, prevent vertical mixing. w θ < 0 gives a little contribution to heat the earth surface, more turbulence would prevent surface freezing at night.

7 Governing equations for Atmospheric Turbulent Flow (1) Note: these equations are valid at any point of the fluid domain

8 Governing equations for Atmospheric Turbulent Flow (2) This set of equations is for a rigorous Navier Stokes simulation, with mass, momentum, heat, moisture conservation equations coupled with the equation of state (all valid at any point x,y,z,t of the flow spatiotemporal domain)

9 1) Ideal gas Law and the thermodynamics of moist air p = ρ air R D T v Air density has to be for moist air R is the dry air gas constant = 287 J/Kg K T v is the virtual absolute temperature for unsaturated air T v = T(1+0.61r) with r = mixing ratio = m wat m dry = ρ wat ρ dry, Note that T v accounts for humidity but it is not adjusted for pressure as the potential temperature θ. This is obvious as pressure enters in the equation directly. mixing ratio r is related to q = specific humidity q = m wat = ρ wat, m tot ρ tot r q So q= or r= r+1 1 q q = specific humidity is different from the relative humidity RH usually estimated in %. RH quantifies how far the moist air is close to saturation (RH=100%) How do we calculate q? q = m wat m tot = ρ wat ρ tot, RH = e w e w sat where e w is the vapor pressure

10 e w sat is a measure of how much water (in the gas state) can be contained in volume of air at a given temperature. There are formulas defining the water vapor pressure at saturation (Buck 1982, Goff Gratch 1957 next slide ). See also Once e w sat is known, we can calculate e w =RH(measured)* e w sat (estimated) Then use q = e w p e w (1 ) where =0.622 = / = m water m dry air the ratio between unit mass of dry air and water respectively Or use directly the mixing ratio r = e w p e w (Be careful not to mix r and q!!!!) Calculation of moist air density ρ = ρ d + ρ w = p e w RT Where R is the (dry) air gas constant = 287 J/Kg K + εe w RT = p RT 1 e w(1 ε) p

11 water vapor pressure at saturation At higher temperature e w sat is larger, so for the same RH e w increases, as well as q

12 Vapor Pressure (at saturation) Analogy: boiling temperature Boiling temperature = temperature at which a liquid boils Below the boiling temperature, the fluid is liquid Boiling temperature is dependent on pressure Vapor pressure (in equilibrium) Vapor pressure = pressure at which a liquid vaporizes Above the vapor pressure, the fluid is liquid Vapor pressure is dependent on temperature

13 continuity equation conservation of mass in an infinitesimal volume dxdydz each mass flux component can be written as ρu da with da perpendicular to u

14 along each direction we have the net mass flux out of the volume: x y z the sum of all the net mass fluxes must be balanced by the variation of mass inside the volume: Thus we can write: Which can be expressed: or

15 2) Continuity equation Note that Dφ = φ + u Dt t j φ x j Lagrangian or material derivative Eulerian derivative + advection of spatial derivative Note that variations in air density depend on temperature, pressure and on the mixing ratio. In the governing equations the (dry) air and the water vapor are treated independently as two components of a gas mixture. Both are uncompressible! yes, but density could be varying due to water vapor mixing ratio...? Think about it... Yes, that is the coupling term! Note that air obeys to the equation of state whereas for liquid water the equation of state is ρ = const For a single fluid system, e.g. dry air we use incompressible fluid equations. However compressibility effects arise when i) pressure affect the density at constant temperature for a single fluid, e.g. for shock waves, or ii) when we look at variations of temperature over mesoscale motions BB week 4.1

16 3) Momentum equation momentum balance in an infinitesimal volume dxdydz Newton second law: Σ F= m a 1) body forces (weight, Coriolis) 2) surface forces (pressure and viscous stresses) 3) net flux of momentum across the volume note that we did not perform any Reynolds decomposition yet, so there are no turbulent stresses

17 body force

18 Σ F= m a 0) eulerian and convective acceleration 1) 2) 0,1,2) =0 for continuity

19 m a = Σ F let us consider the x direction note that m=ρ dv= ρ dxdydz dividing both RHS and LHS of Σ F= m a by dxdydz : along x direction in tensorial notations, for incompressible fluids, dividing by ρ(constant):

3) Coriolis force =-2Ω x u Ω j is the j component of the angular velocity vector of earth rotation, Ω=[0, ω cos(ϕ), ωsin(ϕ)] where 1) ω is the angular velocity of the earth =2π /24hr=7.27 10-5 s.

20 3) Coriolis force =-2Ω x u Ω j is the j component of the angular velocity vector of earth rotation, Ω=[0, ω cos(ϕ), ωsin(ϕ)] where 1) ω is the angular velocity of the earth =2π /24hr= s. (very small, locally it is not important, but averaged over a very large air mass it provides a coherent motion) 2) ϕ is the latitude let us consider i=1 (Eastbound) - 2 ε ijk Ω j u k = - 2 Ω 2 u Ω 3 u 2 = -2 u 3 ω cos(ϕ) + 2 u 2 ωsin(ϕ) at the equator ϕ = 0 : there is a max effects on vertical velocity being tilted in the negative streamwise (West) direction (but vertical velocity is small) at the pole ϕ = π/2 : there is a max effects on u 2 (North) velocity being tilted in the positive streamwise (East)direction (more important effect)

ω http://www.youtube.com/watch?v=qfdqekayvag top view B u2 A u1 EAST earth moving to the east L : low pressure regions attracting air masses from (B) N & (A) S.

21 ω top view B u2 A u1 EAST earth moving to the east L : low pressure regions attracting air masses from (B) N & (A) S. The air masses moving S > N, in a Lagrangian perspective, tilt to the right (E) as their initial E-bound velocity is larger than the E bound velocity of a northern point on the air surface (air masses are not attached to the surface! So they do not lose kinetic energy ) Ω=[0, ω cos(ϕ), ωsin(ϕ)] A: i=1, u=u 2 > - 2 ε ijk Ω j u k = - 2 Ω 2 u Ω 3 u 2 = 2 u 2 ωsin(ϕ) (along i=1 east) B: i=1 u=-u 2 > - 2 ε ijk Ω j u k = - 2 Ω 2 u Ω 3 u 2 = -2 u 2 ωsin(ϕ) (along i=1 west) often the Coriolis term is written as : f c ε ij3 U j with f c =2ωsin(ϕ), so (e.g for i=1 ) neglecting the contribution from the vertical component -2 u 3 ω cos(ϕ)

22 Typhoon Winnie as it approached the Mariana Islands in the Northern Hemisphere on August 13, Source: NASA. Storm in the Southern Hemisphere, east of New Zealand, on December 2, Source: NASA. BB week 4.2

23 Conservation of moisture T T T T net moisture source sink term molecular diffusivity for water vapor in air where q T is the total specific humidity of air (the mass of water per unit mass of moist air). Remember we defined specific humidity q = m wat = ρ wat m tot ρ tot We could divide the specific humidity into 2 contributions: q T = q (vapor) + q L (liquid). We would have two equations: the only differences are 1) in the presence (vapor) or absence (liquid) of the diffusive term. Obviously, water droplets do not diffuse in air 2) in the presence of a source term in the equation for q (+E/ρ) and a sink term in the equation for q L (-E/ρ), where E represent the conversion of liquid water into vapor.

24 Conservation of heat heat source sink term specific heat for moist air= Cp dry (1+0.84q) thermal diffusivity [ J (Kg K) -1 ] where θ is the potential temperature ν θ is the thermal diffusivity S θ = Q j x j EL P Q j is the component of net radiation (from all kind of sources) in the j direction: A gradient of net radiation through our dxdydz control volume induce a change in temperature coupling heat and moisture eqs. Q high θ increase where L P is the latent heat associated to a phase change of E (liquid to vapor or viceversa) L P = J/Kg wat (changing phase) at 0 0 C (function of temperature). EL P can be a source or a sink (for latent heat release) depending on the phase change for sublimation (solid>vap) we have different values Q low Q j x j > 0

25 IMPORTANT: simplifications and approximations 1 1) compressibility 1 ρ dρ dt U j x j true! if the velocity is not very high (shock waves) or if the motion length scale (especially in the vertical direction) are not very large (mesoscale motion) So, yes technically humid air experience density variation, but only over large scales. At the small scales, incompressibility is assumed the flow is assumed to be divergence free, but a correction is required in the vertical momentum equation through a buoyancy term: this is the Boussinesq approximation

2) shallow motion approximation, i.e. the assumptions required for the Boussinesq approximation to be valid (Mahrt 1986) 1) i.e. we can neglect density variation in a relatively shallow layer 0 2 )if.

26 2) shallow motion approximation, i.e. the assumptions required for the Boussinesq approximation to be valid (Mahrt 1986) 1) i.e. we can neglect density variation in a relatively shallow layer 0 2 )if... is balanced it means that the density is constant remember the lapse rate θ=t+(g/c p )z where g/c p ~ K/m d θ/dz~0.01 < K/m (excludes highly convective regime) Spiegel and Voronis (1960): negligible perturbations of thermodynamic variables 3) this suggests that the equation of state can be used to determine the mean properties of the fluid, while it does not have to be coupled with the other equations to solve turbulent motions.

27 following shallow motion approx. linearized perturbation of the ideal gas law, neglect pressure fluctuations This means that density fluctuations are only due to temperature fluctuations (and the latter are measurable!).

28 same order of magnitude, according to assumption 5 Let us focus on the Momentum equation (i=3, U 3 =W) Reynolds decomposition divide by mean density time averaging further neglect subsidence applicable if the shallow convection conditions 1..5 are met

29 IMPORTANT: simplifications and approximations 2

Boussinesq assumption in the 3D momentum equation Simplified momentum equation, Boussinesq approx Boussinesque implies that ρ = ρ & g is replaced by g gθ v /θ v note that θ v is an instantaneous

30 Boussinesq assumption in the 3D momentum equation Simplified momentum equation, Boussinesq approx Boussinesque implies that ρ = ρ & g is replaced by g gθ v /θ v note that θ v is an instantaneous temperature, (fluctuating quantity). The coupling between heat and momentum remains ON. This equation is still equivalent to a DNS This means that the variations in density only enter through the buoyancy term!!! we can work with a mean density defined by the ideal gas law but avoid the coupling between turbulent fluctuations of pressure and density fluctuations. So we can work with incompressible flow equations Now the question is : why Shallow motion approximation? Let us read Mahrt 1986 paper...

1 Introduction to Governing Equations 2 1a Methodology... 2

Contents 1 Introduction to Governing Equations 2 1a Methodology............................ 2 2 Equation of State 2 2a Mean and Turbulent Parts...................... 3 2b Reynolds Averaging.........................