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

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selfe users trainng course 1 selfe physical formulation joseph zhang cmop ohsu
-SELFE Users Trainng Course-(1)SELFE physical formulationJoseph Zhang, CMOP, OHSU
governing equations reynolds averged navier stokes

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
continuity equation 1
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

continuity equation 2
Continuity equation (2)

Vertical boundary conditions: kinematic

z

surface

bottom

Integrated continuity equation

momentum equation 1

>

Momentum equation (1)

Newton’s 2nd law (Lagrangian)

f

Hydrostatic assumption

Separation of scales

Boussinesq assumption

coriolis

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

coriolis7
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

momentum equation v ertical boundary condition b c

ub

db

Momentum equation: vertical boundary condition (b.c.)

Surface

Bottom

  • Logarithmic law:
  • Reynolds stress:
  • Turbulence closure:
  • Reynolds stress (const.)
  • Drag coefficient:

z=-h

momentum equation 3
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)
transport equation 1

>

Transport equation (1)

Mass conservation: advection-diffusion equation

Fick’s law for diffusive fluxes

Advection

B.c.

Diffusion

transport equation 2
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.

air water exchange
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
solar radiation
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

infrared radiation
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

turbulent fluxes of sensible and latent heat
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).

wind shear stress turbulent flux of momentum
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

inputs to heat momentum exchange model
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

turbulence closure pacanowski and philander 1982
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
turbulence closure mellor yamada galperin 1988
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

turbulence closure mellor yamada galperin 198820
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
turbulence closure umlauf and burchard 2003
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)

turbulence closure umlauf and burchard 200322
Turbulence closure: Umlauf and Burchard (2003)

Constants

Wall proximity function

depends on choice of scheme and stability function

geophysical fluid dynamics in ocean
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

geostrophy

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
surface ekman layer

z

wind

d

EL

interior

Surface Ekman layer
  • EL exists whenever viscosity/shear is important
  • Steady, homogeneous, uniform interior:
  • Solution:
ekman transport
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

barotropic waves

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

phase speed
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
group velocity
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

linearization
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

kelvin waves

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
natural frequency of a stratified system
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

internal waves

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

properties of internal waves

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: