Gravity eaves can exist on either:
- The interface at the top of the upper fluid or
- On the interface between the two fluids
Assumes that the space above the uppermost fluid is filled with:
- A fluid of very low density
- Very large depth
- We can use g instead of g’
- P0 at upper interface is zero
- Referred to as free surface
The gravity waves that form on the free surface and ……… are ……. of one another
- The interface
- Not independent
In the upper fluid and lower fluid the PGF is solely due to:
- Upper fluid:
- The slope of the free surface
- Lower fluid:
- Due to the slope of the interface
- Slope of the free surface
To find the dispersion relation we could:
Proceed by assuming sinusoidal function
The disturbance on the free surface is:
- Larger in amplitude than the disturbance on the interior interface
- The total depth of the fluid and the depth of the lower layer, H is greater than the lower depth H1
- Much smaller in amplitude than the disturbance on the interior interface and is of opposite sign
- So the two interfaces are 180 out of phase
Two-layer, shallow-water fluid there are two distinct modes:
- Baroclinic mode
- Barotropic mode
In a three layer fluid it turns out there are three modes:
- Barotropic mode
- Two baroclinic modes
barotropic mode:
the u-components of velocity are also in phase and are nearly equal
baroclinic mode:
the u-components of velocity are also 180 out of phase
an n-layer fluid will have n modes:
barotropic mode
n-1 baroclinic modes
Infinite number of baroclinic modes:
a fluid with continuous stratification can be expected to have an infinite number of baroclinic modes
discussion of two layer fluid will be important later when discussing:
- (concept of equivalent depth) internal waves
- multi-layered
- continuously stratified fluid
Equivalent depth:
depth that the fluid would have to have in order for an external gravity wave to have the same speed as the baroclinic mode
For a fluid with multiple layers:
- there are multiple baroclinic modes.
- Each mode will have an equivalent depth
- We know the dispersion relation
- Each mode will have an equivalent depth
Fluid with continuous stratification:
- There would be an infinite number of baroclinic modes
- Not all modes will be important
- Some will be very weak
- If we can identify the most important baroclinic modes
- Maybe as few as two or three
- Can find equivalent depths for them
- Can find dispersion relations for these modes
- This is known as normal mode analysis
Why do we study the shallow water gravity waves:
- Because of the concept of equivalent depth
- The atmosphere is a multilayer fluid
- Supports many baroclinic modes of oscillation
- Scales of motion studied in synoptic scale dynamics
- The hydrostatic assumption is valud
- To study the linear gravity waves that are supported by the atmosphere
