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(Abridged) Turbulence in the solar wind is believed to generate an energycascade that is supported primarily by Alfv\'en waves or Alfv\'enicfluctuations at MHD scales and by kinetic Alfv\'en waves (KAWs) at kineticscales $k_\perp \rho_i\gtrsim 1$. Linear Landau damping of KAWs increases withincreasing wavenumber and at some point the damping becomes so strong that theenergy cascade is completely dissipated. A model of the energy cascade processthat includes the effects of linear collisionless damping of KAWs and theassociated compounding of this damping throughout the cascade process is usedto determine the wavenumber where the energy cascade terminates. It is foundthat this wavenumber occurs approximately when $|\gamma/\omega|\simeq 0.25$,where $\omega(k)$ and $\gamma(k)$ are, respectively, the real frequency anddamping rate of KAWs and the ratio $\gamma/\omega$ is evaluated in the limit asthe propagation angle approaches 90 degrees relative to the direction of themean magnetic field.

A Kinetic Alfven wave cascade subject to collisionless damping cannot reach electron scales in the solar wind at 1 AU