Atm + Ocean dynamics: Test 3

Camille and Elin UiB

Governing Equations for internal waves

Continuity: ∂u/∂x+∂v/∂y+∂w/∂z=0​

Momentum X-Dir: ∂u/∂t=​-1po dp/dx

Momentum Y-Dir: ∂v/∂t=​-1po dp/dy

Energy dp/dt + w dp/dz = 0

The simplifications in Internal waves

They are:

-Inviscid

-Non Rotating

-Small amplitude

-Non hydrostatic

Dispersion relation

w²=N² k²+l²/k²+l²+m²

What does a fluid parcel do in an internal wave?

Oscillates

Does w depend on total wavenumber k?

yes, you need a higher k for higher w

What are some general properties of internal waves?

For a given stratification, the direction of the wave number vector determines the frequency.

The phase speed is parallel to lines of constant phase.

Lee waves are created through

Topography with waves: trapped (strong stratification) / radiating waves (weak stratification)

Governing equations for internal waves

X-Direction: 𝜕𝑢/ 𝜕𝑡 + 𝑢 𝜕𝑢/ 𝜕𝑥 + 𝑣 𝜕𝑢 /𝜕𝑦 − 𝑓𝑣 = −1𝜌o 𝜕𝑢 /𝜕𝑥

Y-Direction: 𝜕𝑡 + 𝑢 𝜕𝑣/ 𝜕𝑥 + 𝑣 𝜕𝑣/ 𝜕𝑦 + 𝑓𝑢 = −1𝜌 𝜕𝜌/ 𝜕𝑦

Z-Direction (Hydrostatic); 𝜕𝑃/ 𝜕𝜌 = 𝑔z

Continuity: 𝜕ℎ /𝜕𝑡 + 𝜕 /𝜕𝑥 (ℎ𝑢) +𝜕 /𝜕𝑦 (ℎ𝑣) = 0

has p or theta (and can be barotropic/baroclinic)

In the 2 layer model:

u,v,𝜌 are constant

P is not constant, but the gradient is

P1 = 𝑝_atm + 𝜌(ℎ₁+ ℎ₂+𝑏)