# 4. Shallow Layer Dynamics Flashcards

Shallow-Layer Dynamics

Outline

- 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

Shallow-Layer Dynamics

Diagram

- 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 «_space;1
- horizontal velocity scale, u,v~U

Shallow-Water Dynamics

Governing Equations

-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

Shallow Water Dynamics

Hydrostatic Balance

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

Shallow-Water Dynamics

Boundary Conditions on h

-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

Shallow-Water Dynamics

Linearised Shallow-Water Equations

-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)

Shallow-Water Dynamics

f=0

-differentiating the h equation with respect to t:

∂²h/∂t² - gH∇²h = 0

-c.f. sound waves equivalent wave equation

Shallow-Water Dynamics

f≠0

-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!!

Inertia-Gravity Waves

Derivation

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

Inertia-Gravity Waves

Dispersion Relation

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

Inertia-Gravity Waves

f=0

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

Inertia-Gravity Waves

g=0

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

Kelvin Waves

Outline

- same physics as inertial-gravity waves but different

- consider a coast with v=0 everywhere

Kelvin Waves

Governing Equations

-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

Kelvin Waves

Dispersion Relation

-try wave ansatz with y-structure function to be determined in the Kelvin wave governing equations

=>

ω = ±ck, where c = √[gH]