  We use a detailed numerical model of stimulated thermal Rayleigh scattering
to compute mode instability thresholds in Tm$^{3+}$-doped fiber amplifiers. The
fiber amplifies 2040 nm light using a 790 nm pump. The cross-relaxation process
is strong, permitting power efficiencies of 60%. The predicted instability
thresholds are compared with those in similar Yb$^{3+}$-doped fiber amplifiers
with 976 nm pump and 1060 nm signal, and are found to be higher, even though
the heat load is much higher in Tm-doped amplifiers. The higher threshold in
the Tm-doped fiber is attributed to its longer signal wavelength, and to
stronger gain saturation, due in part to cross-relaxation heating.
