-SELFE Users Trainng Course- (1) SELFE physical formulation Joseph Zhang, CMOP, OHSU

1. -SELFE Users Trainng Course-(1)SELFE physical formulationJoseph Zhang, CMOP, OHSU

2. z shear stratification MSL x East Governing equations: Reynolds-averged Navier-Stokes • Continuity equation • Momentum equations • Vertical b.c.: • Equation of state • Transport of salt and temperature • Turbulence closure: Umlauf and Burchard 2003

3. Continuity equation (1) • Law of mass conservation + incompressibility of water • Vertical integration w v Eulerian form: Dz u Dy Dx y z Lagrangian form: Boussinesq: (Volume conservation) x

4. Continuity equation (2) Vertical boundary conditions: kinematic z surface bottom Integrated continuity equation

5. > Momentum equation (1) Newton’s 2nd law (Lagrangian) f Hydrostatic assumption Separation of scales Boussinesq assumption

6. u J j i Wxr I W Coriolis • Coriolis: “virtual force” • Newton’s 2nd law is only validin an inertial frame • For large scale GFD flows, the earth is not an inertial frame • Small as it is, it is reponsible for a number of unique phenomena in GFD • Accelerations in a rotating frame (W const.): Coriolis Centrifugal acceleration

7. Coriolis • Plane coordinates: neglect curvature of the earth • x: east; y: north; z: vertical upward • Projection of earth rotation in the local frame: • Components of Coriolis acceleration: • Inertial oscillation: no external forces, advection and vertical vel. • e.g., sudden shut-down of wind in ocean. y z x ) f W

8. ub db Momentum equation: vertical boundary condition (b.c.) Surface Bottom • Logarithmic law: • Reynolds stress: • Turbulence closure: • Reynolds stress (const.) • Drag coefficient: z=-h

9. Momentum equation (3) meters • Implications • Use of z0 seems most natural option, but over-estimates CD in shallow area • Strictly speaking CD should vary with bottom layer thickness (i.e. vertical grid) • Different from 2D, 3D results of elevation depend on vertical grid (with constant CD)

10. > Transport equation (1) Mass conservation: advection-diffusion equation Fick’s law for diffusive fluxes Advection B.c. Diffusion

11. Transport equation (2) Precipitation and evaporation model E: evaporation rate (kg/m2/s) (either measured or calculated) P: Precipitation rate (kg/m2/s) (usually measured) Heat exchange model IR’s are down/upwelling infrared (LW) radiation at surface S is the turbulent flux of sensible heat (upwelling) E is the turbulent flux of latent heat (upwelling) SW is net downward solar radiation at surface The solar radiation is penetrative (body force) with attenuation (which depends upon turbidity) acts as a heat source within the water.

12. Air-water exchange • free surface height: variations in atmospheric pressure over the domain have a direct impact upon free surface height; set up due to wind stresses • momentum: near-surface winds apply wind-stress on surface (influences advection, location of density fronts) • heat: various components of heat fuxes (dependent upon many variables) determine surface heat budget • shortwave radiation (solar) - penetrative • longwave radiation (infrared) • sensible heat flux (direct transfer of heat) • latent heat flux (heating/cooling associated with condensation/evaporation) • water: evaporation, condensation, precipitation act as sources/sinks of fresh water

13. Solar radiation Downwelling SW at the surface is forecast in NWP models - a function of time of year, time of day, weather conditions, latitude, etc Upwelling SW is a simple function of downwelling SW ais the albedo - typically depends on solar zenith angle and sea state Attenuation of SW radiation in the water column is a function of turbidity and depth d (Jerlov, 1968, 1976; Paulson and Clayson, 1977): R, d1 and d2 depend on water type; D is the distance from F.S. III II Depth (m) I SW

14. Infrared radiation • Downwelling IR at the surface is forecast in NWP models - a function of air temperature, cloud cover, humidity, etc • Upwelling IR is can be approximated as either a broadband measurement solely within the IR wavelengths (i.e., 4-50 μm), or more commonly the blackbody radiative flux eis the emissivity, ~1 sis the Stefan-Boltzmann constant Tsfcis the surface temperature

15. Turbulent Fluxes of Sensible and Latent Heat • In general, turbulent fluxes are a function of: • • Tsfc,, Tair • • near-surface wind speed • • surface atmospheric pressure • • near-surface humidity • Scales of motion responsible for these heat fluxes are much smaller than can • be resolved by any operational model - they must be parameterized (i.e., bulk aerodynamic formulation) where: u*is the friction scaling velocity T*is the temperature scaling parameter q*is the specific humidity scaling parameter rais the surface air density Cpais the specific heat of air Leis the latent heat of vaporization The scaling parameters are defined using Monin-Obukhov similarity theory, and must be solved for iteratively (i.e., Zeng et al., 1998).

16. Wind Shear Stress: Turbulent Flux of Momentum Calculation of shear stress follows naturally from calculation of turbulent heat fluxes Total shear stress of atmosphere upon surface: Alternatively, Pond and Picard’s formulation can be used as a simpler option

17. Inputs to heat/momentum exchange model • Forecasts of uwvw @ 10m, PA @ MSL,Tair and qair @2m • used by the bulk aerodynamic model to calculate the scaling parameters • Forecasts of downward IR and SW Atmospheric properties Heat fluxes

18. Turbulence closure: Pacanowski and Philander (1982) • Kelvin-Helmholtz instability: occurs at interface of two fluids or fluids of different density (stratification) • When stratification is present, the onset of instability is govern by the Richardson number • Theory predicts that the fluid is unstable when Ri<0.25

19. Turbulence closure: Mellor-Yamada-Galperin (1988) • Theorize that the turbulence mixing is related to the growth and dissipation of turbulence kinetic energy (k) and Monin-Obukhov mixing length (l) • k and l are derived from the 2nd moment turbulence theory Dissipation rate Shear & buoyancyfreq Wall proximity function Viscosity and diffusivities Stability functions Galperin clipping

20. Turbulence closure: Mellor-Yamada-Galperin (1988) • Vertical b.c. • TKE related to stresses • Mixing length related to the distance from the “walls” (note the singularity) • Initial condition

21. Turbulence closure: Umlauf and Burchard (2003) • Use a generic length-scale variable to represent various closure schemes • Mellor-Yamada-Galperin (with modification to wall-proximity function  k-kl) • k-e (Rodi) • k-w (Wilcox) • KPP (Large et al.; Durski et al) Wall function Diffusivity Stability: Kantha and Clayson (smoothes Galperin’s stability function as it approaches max)

22. Turbulence closure: Umlauf and Burchard (2003) Constants Wall proximity function depends on choice of scheme and stability function

23. Geophysical fluid dynamics in ocean • Coriolis plays an important role in GFD • Although Ek<<1, the friction term is important in boundary layer, and is the highest-order term in the eq. • Reynolds number: • Eddy viscosity >> molecular viscosity Ekman and Rossby numbers

24. low > > > > high high < < < low W Geostrophy • Rapidly rotating (Coriolis dominant), homogeneous inviscid fluids: • N-S eq: • Taylor-Proudman theorem (vertical rigidity): • Flow is along isobars, i.e., streamline = isobar • No work done by pressure; no energy required to sustain the flow (inertial) • Direction along the isobars

25. z wind d EL interior Surface Ekman layer • EL exists whenever viscosity/shear is important • Steady, homogeneous, uniform interior: • Solution:

26. Ekman transport • Properties: • Velocities can be very large for nearly inviscid fluid • Wind-generated vel. makes a 45o angle to the wind direction • Ekman spiral • Net horizontal transport: • Ekman transport is perpendicular to the direction of wind stress (to the right in the Northern Hemisphere) • Bottom Ekman layer wind z Surface current spiral Ekman ) 45o wind Surface current Ekman transport

27. crest l trough Barotropic waves • Linear wave theory • 2D monochromatic wave: • Fixed (x,y), varying t: wT=2p T=2p/w (period) y t fixed L2 k (x,y) fixed L1 l T x h=0 h=-A h=0 h=A h=A

28. Phase speed • Vary (x,y,t), but follow a phase line L: lx+my-wt=C. • The speed ofa phase • In many dynamic systems, w and k are related in the dispersion relation: e.g., surface gravity waves: • Dispersion: a group of waves with different wavelength travel with different speed • Non-dispersive waves: tides

29. Group velocity • Wave energy ~ A2 • Two 1D waves of equal amplitude and almost equal wave numbers: • The speed at which the envelop/energy travels is the group velocity C Cg l’ average wave Modulation/envelop of average wave

30. Linearization • Linearization of advective terms: Ro << 1 • Shallow water model: • For flat bottom, h=h+H (H: mean depth). Linearizing for small-amplitude waves h~A<<H leads to: • (a-c) govern the dynamics of linear barotropic waves on a flat bottom • Neglecting Coriolis, the 1D wave equation is recovered z h(x,y,t) b(x,y) O (wave continuity equation) phase speed

31. W y u=0 x Kelvin waves • Semi-infinite domain of flat bottom, free slip at coast • (a-c) become (3 eqs. for 2 unknowns; u=0): • Solution: where F is an arbitrary function • Properties: • Kelvin wave is trapped near the coast; Rossby radius of deformation R is a measurement of trapping; • The wave travels alongshore w/o deformation, at the speed of surface gravity waves c, with the coast on its right in the N-H; • The direction of the alongshore current (v) is arbitrary: • Upwelling wave h>0  v<0, same direction as phase speed; • Downwelling wave h<0  v>0, opposite direction to phase speed • As f0, Kelvin wave degenerates to plane gravity wave; • Generated by ocean tides and wind near coastal areas

32. Natural frequency of a stratified system • A simple 1D model for continuously stratified fluid • Horizontally homogenous: r(z) • Static equilibrium initially • Wave motion due to buoyancy • The freq. of oscillation is • If N2 > 0, stable stratification  internal wave motion possible • If N2 < 0, unstable stratification  small disturbances leads to overturning; no internal waves possible r(z+h) z r(z) r(z-h) Brunt-Vaisala frequency

33. Transport eq. Internal waves • Surface & internal gravity waves: same origin (interface); interchange of kinetic and potential energy • Quintessential internal waves: • Over an infinite domain (even in the vertical!) withvertical stratification • No ambient rotation (gravity waves) • Neglect viscosity; small-amplitude motion • Hydrodynamic effects are important • Perturbation expansion: • Wave solution ~ exp[i(lx+my+nz-wt)]  (u,v,w), generated by perturbation, are small

34. k ) q k, C ) q Cg z x Properties of internal waves • For a given N, wmax=N • Dispersion relation only depends on the angle between the wave number and horizontal plane: • For a given frequency, all waves propagate at a a fixed angle to the horizontal • As w0, q90o • Wave motion: • Transverse wave • Group vel. inside the phase plane • Purely vertical motion when n=0; Cg=0  no energy radiation • Purely horizontal motion when l=0; w=0 steady state; blocking • For continuously stratified fluid on a finite depth, the phase speed is bounded by a maximum but there are infinitely many modes of internal waves • Inertial-internal waves: