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

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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 formulationJoseph 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
• Turbulence closure: Umlauf and Burchard 2003
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)

Vertical boundary conditions: kinematic

z

surface

bottom

Integrated continuity equation

>

Momentum equation (1)

Newton’s 2nd law (Lagrangian)

f

Hydrostatic assumption

Separation of scales

Boussinesq assumption

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

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

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)

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)

Fick’s law for diffusive fluxes

B.c.

Diffusion

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
• 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
• 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

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

• 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
• 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

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
• 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)
• 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
• 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

• 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)
• 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 (2003)

Constants

Wall proximity function

depends on choice of scheme and stability function

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

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

z

wind

d

EL

interior

Surface Ekman layer
• EL exists whenever viscosity/shear is important
• Solution:
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

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
• 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
• 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 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

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
• 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

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

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: