# Formula for numerical model for sustained warming in marine sediment?

I have been reading this paper and found simple numerical formulations to model the time duration, affected depth and amount of warming of marine sediments. The author uses a rudimentary estimate of the depth to which sediments are affected by an instantaneous, sustained temperature change DT in the overlying air or ocean waters that can be made using the diffusive length scale:

1 = √kt    (1)


which describes the depth (m) that 0.5 DT will propagate in elapsed time t (s). k denotes thermal diffusivity, which ranges from ~0.6 to 1x10-6 m2/s for unconsolidated sediments.

For instance, it is said that over 10, 100, and 1000 yr, the calculation yields maximum of 18 m, 56 m, and 178 m, respectively, regardless of the magnitude of DT. However, in page 6 (section 3), it is said that using a DT=1.25ºC (plus some other assumptions) during 100 yr of sustained warming affects a depth of ~40 m. How about if just DT is changed to 4ºC? deeper than 40 m in the same 100 yr?

I understand that (1) can be used to estimate both the time of the sustained warming and the affected depth. However, this formula does not provide a way to define the amount of warming or to estimate how much warming was applied if time and depth are known. Is there a formula for the amount of warming that complements (1)?