<p>I keep hearing that the models used for global ocean simulations are not actually dissipative, even though they clearly have dissipative terms and we couldn't solve them if they didn't (here, &quot;solve&quot; means they don't blow up numerically).  I don't find this obvious in the least, and I have not seen this mentioned or discussed in the relevant literature/textbooks (but then I have not spent the hours digging through the literature to find such an explanation, though I anticipate it would take longer to comb the literature than to take a global ocean model and figure it out directly).  The primitive equations are clearly dissipative with a finite-dimensional attractor (moreover one has to artificially inflate the diffusion coefficients to make the simulations run in the first place), so if one tries to solve these equations for a simple model, without implementing any parameterizations, it should simply be dissipative.  Numerical discretization in space definitely creates new issues, but in general such approximations <em>add</em> spurious dissipation.  Any insight into the non-dissipativity of large-scale climate models (with the parameterizations turned off) would be greatly appreciated.</p>

CC BY-SA 4.0
2023-12-23T20:55:34.097Z
My answer to my own question (which I thought about in a clearer frame of mind the next day) was deleted without explanation after seeking explanation, so I am reposting it here in the comments because this answer is useful for me and I hope is useful for others (otherwise I would delete the question).
mathishard.butweloveit
CC BY-SA 4.0
2023-12-23T20:55:43.727Z
The difference between the primitive equations and the equations used in large-scale ocean models is that there is a layer-thickness equation that is evolved in time. This equation is not dissipative. This disqualifies the system as being, strictly, dissipative. It is not immediately clear if the large-time behavior is finite-dimensional or not.
mathishard.butweloveit
CC BY-SA 4.0
2023-12-23T20:55:45.917Z
This led me to reflect on a system of equations that is somewhat analogous (really only in the fact that it has an equation that is non-dissipative), the Boussinesq equations without thermal diffusion. This paper, https://www.sciencedirect.com/science/article/pii/S0294144915300111, shows the existence of a weak attractor for this system which contains infinite-dimensional subspaces. I suspect this is also the case with the global ocean equations, as usually finite dimensional large-time behavior only holds for infinite dimensional systems with dissipation (sometimes damping).
mathishard.butweloveit
<climate-models><numerical-modelling><fluid-dynamics>
Global Ocean Models not dissipative?
mathishard.butweloveit
# Global Ocean Models not dissipative?

(Asked by: username_0 on 2023-12-12)

I keep hearing that the models used for global ocean simulations are not actually dissipative, even though they clearly have dissipative terms and we couldn't solve them if they didn't (here, "solve" means they don't blow up numerically). I don't find this obvious in the least, and I have not seen this mentioned or discussed in the relevant literature/textbooks (but then I have not spent the hours digging through the literature to find such an explanation, though I anticipate it would take longer to comb the literature than to take a global ocean model and figure it out directly). The primitive equations are clearly dissipative with a finite-dimensional attractor (moreover one has to artificially inflate the diffusion coefficients to make the simulations run in the first place), so if one tries to solve these equations for a simple model, without implementing any parameterizations, it should simply be dissipative. Numerical discretization in space definitely creates new issues, but in general such approximations *add* spurious dissipation. Any insight into the non-dissipativity of large-scale climate models (with the parameterizations turned off) would be greatly appreciated.

**Tags:** climate-models, numerical-modelling, fluid-dynamics

> **Comments:**
> **username_0**: My answer to my own question (which I thought about in a clearer frame of mind the next day) was deleted without explanation after seeking explanation, so I am reposting it here in the comments because this answer is useful for me and I hope is useful for others (otherwise I would delete the question).  
> **username_0**: The difference between the primitive equations and the equations used in large-scale ocean models is that there is a layer-thickness equation that is evolved in time. This equation is not dissipative. This disqualifies the system as being, strictly, dissipative. It is not immediately clear if the large-time behavior is finite-dimensional or not.  
> **username_0**: This led me to reflect on a system of equations that is somewhat analogous (really only in the fact that it has an equation that is non-dissipative), the Boussinesq equations without thermal diffusion. This paper, https://www.sciencedirect.com/science/article/pii/S0294144915300111, shows the existence of a weak attractor for this system which contains infinite-dimensional subspaces. I suspect this is also the case with the global ocean equations, as usually finite dimensional large-time behavior only holds for infinite dimensional systems with dissipation (sometimes damping).  

---

## 0 answers

