() ∂ DDt H ∂ ∂∂ ( uH ) ∂∂ Δ − ∂ h + H DuDt u * g * = = + g + u = g = 0 + g = 0 ∂ t t ∂ x x ∂ x € € € €

(1)

Schär, ETH Zürich

Dynamics and Numerics of Shallow Water Flows

Outline:

• governing equations

• dimensionless parameters

• wave propagation, Tsunamis

• hydraulic jumps

• vortex shedding

• Seiche waves

• numerical implementation Christoph Schär

ETH Zürich schaer@env.ethz.ch Supplement to Lecture Notes

“Numerical Modeling of Weather and Climate”

March 2007

2

Shallow water elquations

h

H ρ

ρ

_u

< ρ u

€

Du

Dt +g^*∂

(

h+H

)

∂x =0

€

D Dt= ∂

∂t+u ∂

∂x

€

∂H

∂t +∂(uH)

∂x =0 System of equations

mit

Reduced gravity g*=gΔρ

=gρ − ρ_u

Approximations

• Horizontal velocity u is

independent of height, i.e. u=u(x)

• The influence of the overlaying layer of fluid (e.g. air) can be neglected.

(2)

Schär, ETH Zürich

Phase speed of shallow-water waves

h

H ρ

ρ

_u

< ρ u

€

c= g*H

Phase speed (non-rotating system, f=0)

Tsunami (g* = g = 10 m/s^–2)

H [m] c [m/s] c [km/h]

1 3.2 12

10 10 36

100 32 115

1000 100 360

2000 140 504

4000 200 720

6000 245 882

(non-dispersive)

€

g*=gΔρ

ρ =gρ − ρ_u

ρ (reduced gravity)

Froude number

€

Fr= u g^*H

=advection velocity wave velocity Fr < 1: subcritical

Fr > 1: supercritical

Schär, ETH Zürich

4

Tsunami of December 26, 2004

(3)

Schär, ETH Zürich

Banda Aceh, Sumatra

6

Hydraulic jumps

hydraulic jump

subcritical Fr < 1 supercritcal

Fr > 1 z

y x

Mit zunehmender Distanz vom Auftreffpunkt nimmt die Geschwindigkeit aus Gründen der Massenerhaltung ab. Dies provoziert den Übergang von superkritischer zu subkritischer Strömung.

v

Fr= u

g^*H =advection velocity wave velocity

(4)

Schär, ETH Zürich

Shallow-water flow past a ridge

Velocity Shallow water surface

Topography

Downstream propagating hydr. jump Supercritical

flow

Schär, ETH Zürich

9

Atmospheric flow past a ridge

Contours:

theta (density) Color:

vertical motion

Downstream propagating

feature Vertically

propagating gravity wave

Shooting flow

(5)

Schär, ETH Zürich

SW flow past a ridge: Regime diagram

€

Fr = u _o H _o

€

M = h

_top

H

_o

Fr

M

(Houghton and Kasahara 1968)

12

Tidal Bore

(6)

Schär, ETH Zürich

Island Socorro (Mexiko)

Schär, ETH Zürich

16

Shallow-water flow past an isolated mountain

(7)

Schär, ETH Zürich

Acqua Alta, Venice

31

Typical synoptic situation for Acqua Alta

Sea-Level Pressure Geopotential 500 hPa

November 7, 1999, 06 UTC (Buzzi et al. 2003; MAP IOP 15)

Seiche-Wave driven by:

• pressure (10 hPa = 10 cm)

• wind Heavy

precipitation

L H

L L

L L H

H

Alps

(8)

Schär, ETH Zürich

Periods of Seiche waves

€

P= 2L gH

€

P= L gH

€

P= 4L gH

Adriatic sea:

L=700 km H=200 m P=17h P=4.25h P=8.5h 1st order Seiche wave

2nd order Seiche wave

Open Seiche wave of 1st order

Schär, ETH Zürich

33

Dimensionless formulation of shallow-water equations

h

H ρ

ρ

_u

< ρ u

€

Du

Dt +∂

(

h+H

)

∂x =0

D Dt= ∂

∂t+u ∂

∂x

€

∂H

∂t +∂(uH)

∂x = 0 Dimensionless formulation

with

€

Du

Dt +g^*∂

(

h+H

)

∂x =0

€

∂H

∂t +∂(uH)

∂x =0 Dimensional formulation

(9)

Schär, ETH Zürich

Numerical Implementation

h

H ρ

ρ

_u

< ρ u

Staggered grid

€

Hⁿ_j−1 h_j−1

x

€

H_jⁿ h_j

€

H_jⁿ₊₁ h_j+1

€

uⁿ_{j+1/ 2}

€

uⁿ_j_{−1/ 2}

€

Δx Array structure

h[j-1] h[j] h[j+1]

u[j] u[j+1]

35

Numerical Integration of Momentum Equation

€

Du

Dt +∂

(

h+H

)

∂x =0 Dimensionless formulation

€

1

2Δt

[

uⁿ⁺¹_{j+1/ 2}−uⁿ⁻¹_{j+1/ 2}

]

⁺^u₂ⁿ^{j+1/ 2}_Δx

[

^uⁿ^{j+3 / 2}⁻^uⁿ^{j−1/ 2}

]

⁺_Δx¹

[ (

^Hⁿ^j+1⁺^h^j+1

)

⁻

(

^Hⁿ^j⁺^h^j

) ]

⁼⁰

€

D Dt= ∂

∂t+u ∂ with ∂x

uⁿ_{j+1/ 2}⁺¹ = uⁿ_{j+1/ 2}⁻¹ − uⁿ_{j+1/ 2} Δt

Δx

[

uⁿ_j_{+3 / 2}−uⁿ_{j−1/ 2}

]

⁻^2Δt_Δx

[ (

^Hⁿ^j+1⁺^h^j+1

)

⁻

(

^Hⁿ^j⁺^h^j

) ]

Centered differences in space and time

Solve for time level n+1

(10)

Schär, ETH Zürich

Numerical Integration of Mass Equation

€

∂H

∂t +∂(uH)

∂x =0

Dimensionless formulation

Centered differences in space and time

Solve for time level n+1

€

1

2Δt

[

Hⁿ⁺¹_j −Hⁿ⁻¹_j

]

⁺_2Δx¹

[

^uⁿ^j+1^H^j+1ⁿ ⁻^uⁿ^j−1^H^j−1ⁿ

]

⁼0 with uⁿ_j=1

2

(

uⁿ_{j−1/ 2}+uⁿ_{j+1/ 2}

)

€

H_jⁿ⁺¹ = Hⁿ⁻¹_j − Δt

Δx

[

uⁿ_j+1Hⁿ_j+1−uⁿ_j−1Hⁿ_j−1

]

with uⁿ_j =1

2

(

uⁿ_{j−1/ 2}+uⁿ_{j+1/ 2}

() ∂ DDt H ∂ ∂∂ ( uH ) ∂∂ Δ − ∂ h + H DuDt u * g * = = + g + u = g = 0 + g = 0 ∂ t t ∂ x x ∂ x € € € €

Dynamics and Numerics of Shallow Water Flows

Shallow water elquations

h

H ρ

ρ

< ρ u

(

)

Phase speed of shallow-water waves

h

H ρ

ρ

< ρ u

Tsunami of December 26, 2004

Banda Aceh, Sumatra

Hydraulic jumps

Shallow-water flow past a ridge

Atmospheric flow past a ridge

SW flow past a ridge: Regime diagram

€

Fr = u o H o

€

M = h

H

Fr

M

Tidal Bore

Island Socorro (Mexiko)

Shallow-water flow past an isolated mountain

Acqua Alta, Venice

Typical synoptic situation for Acqua Alta

L H

L L

L L H

H

Periods of Seiche waves

Dimensionless formulation of shallow-water equations

h

H ρ

ρ

< ρ u

(

)

(

)

Numerical Implementation

h

H ρ

ρ

< ρ u

x

Numerical Integration of Momentum Equation

(

)

[

]

[

]

[ (

)

(

) ]

[

]

[ (

)

(

) ]

Numerical Integration of Mass Equation

[

]

[

]

(

)

[

]

(

)

Fr = u _o H _o