4. Shallow Layer Dynamics

Question 1

Shallow-Layer Dynamics

Outline

Answer 1

• simplest to consider coupled gravitational and rotational dynamics in a shallow-layer system (as opposed to stratification)
• assume a single step in density
• e.g. free water surface, in water ρ=const., in air ρ~0

Question 2

Shallow-Layer Dynamics

Diagram

Answer 2

• neutral flat level of water surface is z=0
• rigid, uniform ocean bed at z=-H
• perturbation of the free surface z=h(x,y,t) which allows for |h|~|H|
• ‘wavelength’=L
• assume |h’| ~ H/L &laquo_space;1
• horizontal velocity scale, u,v~U

Question 3

Shallow-Water Dynamics

Governing Equations

Answer 3

-from incompressibility, W ~ HU/L << U so can disregard Dw/Dt
=>
Du/Dt - fv = -1/ρ ∂p/∂x
Dv/Dt + fu = -1/ρ ∂p/∂y
0 = -1/ρ ∂p/∂z - g
∂u/∂x + ∂v/∂y + ∂w/∂z = 0

Question 4

Shallow Water Dynamics

Hydrostatic Balance

Answer 4

p = -ρg(z - h(x,y,t)) + po

• where the constant of integration is such that p=po at z=h
• this determines p everywhere
• can sub this back into the governing equations to eliminate p

Question 5

Shallow-Water Dynamics

Boundary Conditions on h

Answer 5

-the kinematic condition states that fluid elements on the surface remain on the surface:
D/Dt (z-h(x,y,t)) = 0
-this is equivalent to no penetration at the surface
-the same as:
w = Dh/Dt at z=h
i.e.
w(x,h) = Dh/Dt

Question 6

Shallow-Water Dynamics

Linearised Shallow-Water Equations

Answer 6

-linearise governing equations D/Dt derivatives become ∂/∂t
∂u/∂t - fv = -g ∂h/∂x
∂v/∂t + fu = -g ∂h/∂y
∂h/∂t = -H(∂u/∂x + ∂v/∂y)

Question 7

Shallow-Water Dynamics

f=0

Answer 7

-differentiating the h equation with respect to t:
∂²h/∂t² - gH∇²h = 0
-c.f. sound waves equivalent wave equation

Question 8

Shallow-Water Dynamics

f≠0

Answer 8

-introduces a new phenomenon
-this makes the linearised shallow-water equations a neat framework to study rotational-gravitational phenomenon
-if inertial terms are neglected:
-fv = -g ∂h/∂x
fu = -g ∂h/∂y
-the equations of geostrophic balance!!

Question 9

Inertia-Gravity Waves

Derivation

Answer 9

-try a multivariable wave ansatz for u, v and h in the linearised shallow-water equations

Question 10

Inertia-Gravity Waves

Dispersion Relation

Answer 10

ω[ω² - f² - gH(k² + l²)] = 0
-so have 
ω=0, geostrophic balance
OR
ω² = f² - gH(k² + l²)

Question 11

Inertia-Gravity Waves

f=0

Answer 11

f=0
=>
ω = ± √[gH] k
-gravity waves, governing equation:
∂²h/∂t² - gH∇²h = 0
-recovers gravity waves

Question 12

Inertia-Gravity Waves

g=0

Answer 12

g=0 
=>
ω = ±f
-frequency of waves is the same as the Coriolis frequency
-recovers inertial waves

Question 13

Kelvin Waves

Outline

Answer 13

• same physics as inertial-gravity waves but different

- consider a coast with v=0 everywhere

Question 14

Kelvin Waves

Governing Equations

Answer 14

-take linearised shallow-water equations and sub in v=0
=>
-inertial balance:
∂u/∂t = -g ∂h/∂x
-geostrophic balance:
fu = -g ∂h/∂y
∂h/∂t = -H ∂u/∂x
-give two possibilities, evanescent (exponential) and sinusoidal

Question 15

Kelvin Waves

Dispersion Relation

Answer 15

-try wave ansatz with y-structure function to be determined in the Kelvin wave governing equations
=>
ω = ±ck, where c = √[gH]

Question 16

Kelvin Waves

h^

Answer 16

h^ = ∓ fc/g h^’

-since y≥0, need h^ not tending to infinity as y->∞

Question 17

Kelvin Waves

Northern Hemisphere

Answer 17

f = fo > 0
=>
h^ = A exp(∓ fc/g y)
-if ω=-ck, h^ = A exp(+ fc/g y) -> ∞ unavoidably, need to exclude this solution
-if ω=+ck, h^= A exp(- fc/g y) which decays as y->∞
cp = ω/l = c
-waves propagate to the right

Question 18

Kelvin Waves

Southern Hemisphere

Answer 18

f = fo < 0

-waves propagate to the left

Question 19

Kelvin Waves

Equator

Answer 19

fo=0
-so need β-plane
f = βy
-get a different result for h^, a Gaussian not an exponential
-waves propagate east

Question 20

Potential Vorticity

Description

Answer 20

• consider a fluid column in a flow that spins at the same rate it is advected
• shallow-water system

Question 21

Potential Vorticity

Governing Equation

Answer 21

-absolute vorticity:
q = f(y) + ζ
-in shallow-water systems, vortex stretching operates:
Dq/Dt = q_ . ∇ω_ ≠ 0

Question 22

Potential Vorticity

Definition

Answer 22

-although q is not conserved, a related quantity is:
Q = q / [H+h]
-this represents the total angular momentum of a fluid column in a shallow water system
-angular momentum is conserved and Q is materially conserved

Question 23

Rossby Adjustment

Description

Answer 23

• time-dependent adjustment of a shallow-water system from a geostrophic imbalance
• initial disturbance in h say, how does this evolve?

Question 24

Rossby Adjustment

Governing Equations

Answer 24

-focus on linearised shallow-water equations with f=fo
-and the linearised form of DQ/Dt=0
=>
∂/∂t (ζ - hf/H) = 0
-where S(x,t)=ζ-hf/H represents linearised Q
-since ∂S/∂t=0, S is a constant, S=So(x) set by the initial conditions

Question 25

Rossby Adjustment

Final Steady State

Answer 25

-suppose initial conditions h=ho(x), u=uo(x), v=vo(x) in 1D
-consider ∂/∂y=0
-focus on the final steady state, so ∂/∂t=0
=>
d²h/dx² - h/L² = f/g So(x)
-and
L² = gH/f²

Question 26

Wind-Driven Ocean Circulation

Description

Answer 26

• imagine an ocean basin

- model using linearised shallow-water equations and assuming steady flow

Question 27

Wind-Driven Ocean Circulation

Governing Equations

Answer 27

-linearised shallow water equations
-fv = -g ∂h/∂x - ku + F(y)
-where -fv is the Coriolis force, -g ∂h/∂x is gravity, -ku is drag from the ocean bed and F(y) is the wind drive
fu = -g ∂h/∂y - kv
-and mass conservation:
∂u/∂x + ∂v/∂y = 0

Question 28

Wind-Driven Ocean Circulation

k=0

Answer 28

-sub k=0 into governing equations
-need β-plane approximation for f or get ∂F/∂y=0
f = fo + βy
=>
v = -1/β ∂F/∂y
-this implies the wind directly => v
-but for a closed basin, to conserve mass, we have;
∫v dx = 0
-where the integral is across the width of the basin from x=0 to x=a
-a contradiction

Question 29

Wind-Driven Ocean Circulation

k>0

Answer 29

• βv = ∂F/∂y + k (∂v/∂x - ∂u/∂y)
• 2D incompressibility holds so can introduce streamfunction
• anticipate boundary layer to solve ∫v dx = 0 problem

Question 30

Non-Linear Shallow-Water Dynamics

Examples

Answer 30

• tidal bore
• turbidity current -> sediment / particle laden flow in ocean caused e.g. by mudslide
• pyroclastic flow
• avalanches

Question 31

Non-Linear Shallow-Water Dynamics

Governing Equations

Answer 31

-recall the non-linear shallow-water equations:
–mass conservation
∂h/∂t + ∂/∂x[hu] = 0
–momentum conservation
∂u/∂t + u ∂u/∂x = -g ∂h/∂x
-where h is now used to denote the full thickness, not the perturbed height as used for the linearised case

Question 32

Non-Linear Shallow-Water Dynamics

Steady Flow

Answer 32

-for steady flow, ∂/∂x[hu]=0 so:
hu = Q = const.
-and from the governing equations:
1/2 u² + gh = K
-sub in 
=>
1/2 Q²/h² + gh = K

Question 33

Non-Linear Shallow-Water Dynamics

Theory of Shocks

Answer 33

• non-linear shallow water systems naturally steepen
• in many situations (e.g. tidal bores) generating a shock, a region with an effectively vertical, continuously overturning a ‘hydraulic jump’
• turns out we can develop a direct theory for the evolution of xs(t), the shock position

Question 34

Non-Linear Shallow-Water Dynamics

First Rankine-Hugonist Relation

Answer 34

-as ε->0
-xs’[h]± + [hu]± = 0
=>
xs’[h]± = [hu]±
-where ε is a small region either side of the shock front, xs

Question 35

Non-Linear Shallow-Water Dynamics

Second Rankine-Hugonist Relation

Answer 35

-start with the governing equation in the x direction
∂u/∂t + u ∂u/∂x = -g ∂h/∂x
-write in conservative form
-integrate wrt x between xs-ε and xs+ε
=>
xs' [hu]± = [hu² + 1/2 gh²]±

Question 36

Gravity Currents

Description

Answer 36

-a shock coupled to time-dependent evolution of a shallow-water system for 0≤x≤xn(t)

Question 37

Gravity Currents

Governing Equations

Answer 37

-start with the non-linear shallow water equations:
∂h/∂t + ∂/∂x[hu] = 0
∂u/∂t + u ∂u/∂x = -g ∂h/∂x

Question 38

Gravity Currents

Steps

Answer 38

-apply the first Rankine-Hugoniot relation
-apply momentum balance at the front
-integrate from xn-ε to xn+ε
-replace with xn’=un
=>
1/2 u² = gh at x=xn(t)
-the so called Froude-number condition
-treat this as a boundary condition on the nonlinear shallow-water equations

Question 39

Gravity Currents

Froude Number

Answer 39

-the Froude number:
Fr = √[u²/gh]
-at the front, Fr=√2