Classifying Rossby wave as long or short at 35°N with 500 km wavelength

[geojon]

Homework Statement

Consider a Rossby wave propagating in a homogeneous ocean with a water depth of 4,000m at 35^o N latitude. If the wave length of this wave is 500km, is this a long or short Rossby wave?

Homework Equations

shortwaves: |k_x^-1| <<R_D
longwaves: |k_x^-1| >> R_D

k_x is the x-component of the wave number, k. wavenumber = 2pie/λ
-k_x = k

R_D = Rossby Radius of Deformation = (gh)^1/2 / ƒ
where ƒ is the Coriolis parameter.

The Attempt at a Solution

R_D = Rossby Radius of Deformation = (gh)^1/2 / ƒ = 2.38 x 10⁶m. ƒ is in seconds^-1, R_D is in meters.

-k_x = k = 2Pie / λ. = 1.26 x 10^-5 rad m^-1.

I want to compare |k_x^-1| to R_D. How do I change my units for k_x?? Because I cannot compare something in rad*m^-1 to meters.

I think that I have to change the radians to a distance, using the Earth's radius (6,371 km), at a latitude of 35^o North. This is where I am stumped??

 

[jedishrfu]

One radian on the Earth = 1 Earth radius in length right?