  The longitudinal spin Seebeck effect refers to the generation of a spin
current when heat flows across a normal metal/magnetic insulator interface.
Until recently, most explanations of the spin Seebeck effect use the
interfacial temperature difference as the conversion mechanism between heat and
spin fluxes. However, recent theoretical and experimental works claim that a
magnon spin current is generated in the bulk of a magnetic insulator even in
the absence of an interface. This is the so-called intrinsic spin Seebeck
effect. Here, by utilizing a non-local spin Seebeck geometry, we provide
additional evidence that the total magnon spin current in the ferrimagnetic
insulator yttrium iron garnet (YIG) actually contains two distinct terms: one
proportional to the gradient in the magnon chemical potential (pure magnon spin
diffusion), and a second proportional to the gradient in magnon temperature
($\nabla T_m$). We observe two characteristic decay lengths for magnon spin
currents in YIG with distinct temperature dependences: a temperature
independent decay length of ~ 10 ${\mu}$m consistent with earlier measurements
of pure ($\nabla T_m = 0$) magnon spin diffusion, and a longer decay length
ranging from about 20 ${\mu}$m around 250 K and exceeding 80 ${\mu}$m at 10 K.
The coupled spin-heat transport processes are modeled using a finite element
method revealing that the longer range magnon spin current is attributable to
the intrinsic spin Seebeck effect ($\nabla T_m \neq 0$), whose length scale
increases at lower temperatures in agreement with our experimental data.
