EVAT 554 OCEAN-ATMOSPHERE DYNAMICS TIME-DEPENDENT DYNAMICS; WAVE DISTURBANCES LECTURE 21

“Buoyancy Waves ” Recall the vertical momentum balance for a nearly incompressible fluid that is perturbed from its initial state More generally we have travelling wave disturbances... Represents vertical oscillations due to restoring force of gravity, given some initial perturbation (eventually damped by friction)

“Gravity Waves ” Lateral pressure gradients arise from the perturbed free surface: Consider a perturbation from geostrophic balance The continuity equation takes the approximate form (for small h): More generally we have travelling wave disturbances...

“Gravity Waves ” Lateral pressure gradients arise from the perturbed free surface: The continuity equation takes the approximate form (for small h): Consider a perturbation from geostrophic balance

“Gravity Waves ” The continuity equation takes the approximate form (for small h): Assume the scale of motion is small compared to the planetary scale Differentiate these expressions, Consider a perturbation from geostrophic balance

“Gravity Waves ” This is the equation of a traveling wave! For simplicity, assume that the free surface gradient is non-zero only along the x direction The solution to this equation is: Shallow Water waves Dispersion relation

“Gravity Waves ” This is the equation of a traveling wave! For simplicity, assume that the free surface gradient is non-zero only along the x direction The solution to this equation is: Dispersion relation Shallow Water waves

“Gravity Waves ” Shallow Water waves

Planetary (“Rossby”) Waves ” Consider once again a perturbation from geostrophic balance

Relative Vorticity Absolute Vorticity Recall from an earlier lecture Planetary (“Rossby”) Waves ”

This gives an expression for the Vorticity in the absence of any frictional stresses Consider once again a perturbation from geostrophic balance Planetary (“Rossby”) Waves ”

This gives an expression for the Vorticity in the absence of any frictional stresses Consider once again a perturbation from geostrophic balance Define the streamfunction Planetary (“Rossby”) Waves ”

This gives an expression for the Vorticity in the absence of any frictional stresses Conservation of absolute vorticity on a beta plane, gives Define the streamfunction Planetary (“Rossby”) Waves ”

Define the streamfunction Conservation of absolute vorticity on a beta plane, gives linearize under the assumption of a constant zonal flow The solution has the form of a traveling wave: Planetary (“Rossby”) Waves ”

Define the streamfunction The solution has the form of a traveling wave: Plugging the traveling wave solution into the equation gives, Dispersion Relation If the meridional velocity field represents a geostrophically-balanced standing wave perturbation of the free surface Then we have Planetary (“Rossby”) Waves ”

Plugging the traveling wave solution into the equation gives, Dispersion Relation If the meridional velocity field represents a geostrophically-balanced standing wave perturbation of the free surface Then we have Planetary (“Rossby”) Waves ” Rossby Radius If h=1 km, r  1500km

Dispersion Relation Planetary (“Rossby”) Waves ” The periods of Rossby Waves in the Ocean that are possible is Determined by Latitude and Basin Width

“Kelvin Waves ” Lateral pressure gradients arise from the perturbed free surface: Consider again a perturbation from geostrophic balance The continuity equation takes the approximate form (for small h):

“Kelvin Waves ” The continuity equation takes the approximate form (for small h): Do not assume that the scale of motion is small compared to the planetary scale Consider again a perturbation from geostrophic balance Consider an east-west boundary v=0  v/  t =0

“Kelvin Waves ” The solution is

“Kelvin Waves ” The solution is Thus we have

“Kelvin Waves ” This also generalizes to equatorially-trapped waves! These are “coastally-trapped” waves Development would be identical for North-South boundary where u=0 Length scale is Rossby Radius Equatorial Radius of Deformation:

“Kelvin Waves ” The free surface (  ) can be interpreted in terms of the mean depth of the thermocline

“Kelvin Waves ” The free surface (  ) can be interpreted in terms of the mean depth of the thermocline