From this point of view, MEP allows drawing predictions heat fluxes, specific energy, and temperature profiles from the partial knowledge of the radiative budget and the energy content, while the effect of the sub-grid-scale processes is unknown. The gap between the model and observations is easily understood and explains why our 1-D description with constant insolation is more adapted for the tropics than arctic conditions.
Further improvements are needed to solve these problems. Firstly, a more general 2- or 3-dimensional mass scheme transport is required. Finally, we need to include a time dependence in our model for seasonal or diurnal cycles. A long-term objective might be to construct an SCM, with a limited number of adjustable parameters.
We have investigated the possibility of computing the vertical energy fluxes and temperature in the atmosphere using the MEP closure hypothesis into a simple climate model. The fluxes are then computed in an implicit way, which avoids tuning parameters. This paper provides the first attempt, to the best of the authors' knowledge, to introduce such a representation in a MEPM.
Different energy terms can be considered: sensible heat, geopotential, and latent heat for a saturated profile.
Bertrand's Paradox and the Maximum Entropy Principle
We have shown that this better energetic description of convection allows for obtaining more physically relevant temperature, specific energy, and energy flux profiles, still without any adjustable parameter for the dynamics. In particular, considering geopotential leads to stratification in the upper atmosphere and allows us to reproduce a temperature gradient closer to the observed one.
We have investigated the sensitivity of the model when the atmospheric composition is modified. The results were compared to previous MEPMs and the literature. We hope that the present model may be helpful to construct SCMs with a reduced number of adjustable parameters. A Python code, based on the module scipy. One first writes. Then, we can compute the mean elevation of a layer with two possible prescriptions:.
In both cases, for imposed pressure levels, we obtain the following expression of the specific energy,. In order to solve the optimization problem Eq.
Maximum entropy priors
We therefore search the critical points of the Lagrangian associated with this problem. We linearize the radiative budget and specific energy around a given temperature profile. The entropy production and constraints are then quadratic forms of energy fluxes that can be solved numerically. This is a rather standard procedure for optimization though there is no guarantee of finding the global solution in case of multiple local maxima. To overcome this issue, we start with various random initial temperature profiles that may lead to different local maxima.
In the end, we retain only the best maximum, which is assumed to be the maximum of entropy production. If we assume R to be invertible, the energy flux can be computed with. Considering the atmosphere is an ideal gas at hydrostatic equilibrium, and for prescribed pressure levels, layer volume depends only on temperature. If we linearize around X 0 , one obtains. Using Eq. B4 gives the expression of energy as a function of flux. We also can take into account the latent heat for a water-vapour-saturated atmosphere.
The mixing ratio at the saturation point, q s , depends only on temperature T in K and pressure p in Pa. If we linearize around the profile X 0 ,. Then, we can use the same reasoning as for the dry static heat and replace the matrix E d by E s to consider the effect of latent energy for a saturated moisture profile. However, the radiative budget is still computed with reference water vapour profiles.
In the following, e can represent e d or e s. Using the linearized energy budget Eq. B4 and the constraint Eq. B12 , the problem Eq. As N increases, the algorithm converges rapidly to a solution Fig. DP developed the model code. It is not associated with a conference. We thank Stan Schymanski and the anonymous reviewer for their useful comments that helped us to improve the article.
This paper was edited by Michel Crucifix and reviewed by Stan Schymanski and one anonymous referee. Bolton, D. Weather Rev. CO;2 , Boyd, S. Dewar, R.
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Maximum Entropy Methods
Advances in Geosciences. Host deep learning models on a GPU server for free. Get the week's most popular data science research in your inbox - every Saturday. Contribute to this article. This could be treated has a prior for making a bayesian analysis. Notice that we would use some information about the data to define this prior, namely its expected value not much information but still. Maximum Entropy June, MaxEnt Approximation by Optimization We can apply optimization with the necessary constraints to get approximate discrete distribution to the theoritical MaxEnt solutions. This next is the original eg converted to R: We consider a probability distribution on equidistant points in the interval [-1,1].
We impose the following prior assumptions: