1. In the first paragraph, \psi is the angle between k and v.

2. Taylor hypothesis is valid when the Doppler-transformed frequency in the satellite frame is much larger than the wave frequency in the plasma frame.

3. Otherwise the satellite measures the usual Doppler frequency (w - k*v) containing both the wave vector (k) and the wave frequency (w) in the plasma frame.

4. In this last case you need to know which wave mode you observe. Measured polarization ratios can help discriminating wave modes.

5. Wave vectors of kinetic Alfv'en waves (KAWs) are large enough that the Taylor hypothesis is valid. Then the measured power P(f) is an integral power of all waves with k satisfying k*v = f (k*v means the scalar product). Since k_\perp >>k_\parallel , and the turbulence power decreases with k, the measured spectrum P(f) is in fact a 1D k_\perp spectrum of KAWs with k_\perp = (2 \pi f ) / (v sin(\theta)) (i.e., spectrum of waves with k_\perp lying in (v,B) plain). Here \theta is the (v,B) angle and the above is correct if k_\perp sin(\theta) > k_\parallel (i.e. \theta is not close to 0) . \perp (\parallel) means perpendicular (parallel) to B.

First of all, thank you for your elaborate answer.

I'm glad you say, that in case of KAWs with k_\perp sin(\theta) > k_\parallel (assuming that vpl~va) the observed spectrum P(f)~P(k_\perp), because this is exactly the approximation I use in my forthcoming paper. :) I assume the same is right in case of whistler waves?

Whistlers have higher frequencies, so an interference of spatial and time variations is possible (both k and w can contribute to measured f). This should be checked using solar wind parameters in the Doppler-shifted whistler dispersion (which is equal to f).

BTW I joint the RG network today, just accidentally :)