A Numerical Study of Stochastic El Niño Southern Oscillations Using Wiener Chaos Expansion and Monte Carlo Methods

Yusuf Aydogdu; N. Sri Namachchivaya

doi:10.16993/tellusa.4067

A Numerical Study of Stochastic El Niño Southern Oscillations Using Wiener Chaos Expansion and Monte Carlo Methods

Tellus

Volume 76 (2024): Issue 1

By: Yusuf Aydogdu and N. Sri Namachchivaya

Open Access

|Sep 2024

Figures & Tables

Initial conditions for Kelvin and Rossby waves at time $t = 0$ that satisfy the boundary conditions (left), Kelvin wave at time $t = L / 2 c$ (middle) and Rossby wave at time $t = 2 L / c$ (right)

Kelvin and Rossby waves through characteristics.

Mean and variance of Kelvin and Rossby waves over time (left) and at the terminal time $T = 4 L / c$ (right).

Random field solutions of oceanic Kelvin and Rossby waves generated using different stochastic bases.

Table 1

Gaussian random variables multiplied by the corresponding propagator coefficients for different sets.

ξ_P₀	ξ_P₁	ξ_P₂	ξ_P₃	ξ_P₄	ξ_P₅	ξ_P₆	ξ_P₇	ξ_P₈	ξ_P₉
1.0000	–1.3102	–0.0109	1.7551	1.3406	0.7429	–1.2924	1.0150	0.2958	2.0311
1.0000	0.3678	–0.5975	–0.0969	0.0215	–0.3817	–0.7041	0.3456	–0.6991	–0.1101
1.0000	0.7719	–0.6630	1.3424	0.7646	–1.2890	0.1414	–1.5045	–1.1763	–1.2049
1.0000	0.0194	–0.2582	0.6723	0.8448	–1.0832	2.0494	0.4994	1.1276	–2.1716
1.0000	–0.1005	–0.1200	–0.0088	0.8219	0.3810	0.1785	1.5316	–0.3415	–0.6532

Meridional profiles of atmosphere ( $ϕ_{0, 2} (y)$ ) and ocean ( $Ψ_{0, 2} (Y)$ ) parabolic cylinder functions and.

Zonal wind burst profile $s_{p} (x)$ (left) and zonal thermocline feedback profile $η (x)$ (right).

Deterministic and stochastic wind activities (left) and response of the SST to the different atmosphere-ocean coupling $κ$ at the mid-ocean (right).

Variance of OU-process obtained from WCE with different number of modes $σ = 1.2$ , $d_{p} = 1.0$ and $\hat{τ} = - 0.25$ (left) and $L 1$ errors relative to the exact (analytical) solution (right).

Table 2

Numerical scheme for the solution of deterministic propagators and combining with the appropriate random basis to obtain WCE solution of SPDEs.

NUMERICAL ALGORITHM FOR WIENER CHAOS EXPANSION

Choose a two-way of truncation for the (deterministic) propagator equations and random basis functions.
Truncate the number of propagator equations $K \geq 1$ .
Define grid points in the $(t, x) \in [0, T] \times U$ for the appropriate finite difference scheme.
In this problem, we use $Δ t \approx 3.0 \times 10^{- 2}$ and $Δ x \approx 2.4 \times 10^{- 2}$ and simulate the model with $t \in [0, 28.8]$ and $x \in [0, 1.2]$ .
For each time $t_{i} = i Δ t$ and $x_{j} = j (Δ x_{1}, . . ., Δ x_{n})$ , solve the deterministic propagator system of equations.
From the propagator solutions, construct the statistical moments, i.e., mean and variance.
(optional) Generate random variable $ξ_{k}$ , $k = 1, 2, . . .$ and compute the random field solution.
$\begin{array}{l} K^{A} (t, x) & = \sum_{α \in J} K_{α}^{A} (t, x) V_{α} (ξ); R^{A} (t, x) = \sum_{α \in J} R_{α}^{A} (t, x) V_{α} (ξ); T (t, x) = \sum_{α \in J} T_{α} (t, x) V_{α} (ξ) \\ [5 p t] K^{O} (t, x) & = \sum_{α \in J} K_{α}^{O} (t, x) V_{α} (ξ); R^{O} (t, x) = \sum_{α \in J} R_{α}^{O} (t, x) V_{α} (ξ); τ (t) = \sum_{α \in J} τ_{α} (t) V_{α} (ξ) \end{array}$

Table 3

Comparison of relative mean and variance errors for MC ensembles and WCE method for sea surface temperature $T (t, x)$ , including computation times on a common laptop.

RELATIVE ERROR COMPARISON: MC ENSEMBLES VS. WCE
METHOD	MEAN ERROR	COMPUTATION TIME	VARIANCE ERROR
10 MC Ensembles	0.3792	168 sec	0.4150
50 MC Ensembles	0.0853	798 sec	0.1241
100 MC Ensembles	0.0645	1627 sec	0.1066
300 MC Ensembles	0.0635	4880 sec	0.0616
50 WCE Modes	–	801 sec	–

Comparison of WCE and MC variance results of $T (t, x)$ with different number of ensembles at a fixed space point $x = 0.6$ .

(a) Mean of oceanic Kelvin wave $K^{O} (t, x)$ obtained from WCE (1st propagator), MC (300 ensembles), and the corresponding $L 1$ error. (b) Variance of oceanic Kelvin wave obtained from WCE (50 propagators), MC (300 ensembles).

(a) Mean of oceanic Rossby wave $R^{O} (t, x)$ obtained from WCE (1st propagator), MC (300 ensembles), and the corresponding $L 1$ error. (b) Variance of oceanic Rossby wave obtained from WCE (50 propagators), MC (300 ensembles).

(a) Mean of sea surface temperature (SST) $T (t, x)$ obtained from WCE (1st propagator), MC (300 ensembles), and the corresponding $L 1$ error. (b) Variance of sea surface temperature (SST) obtained from WCE (50 propagators), MC (300 ensembles).

References

Authors

Metrics

Articles in this issue

DOI: https://doi.org/10.16993/tellusa.4067 | Journal eISSN: 3035-9554

Journal RSS Feed

Language: English

Page range: 193 - 205

Submitted on: Mar 26, 2024

Accepted on: Aug 9, 2024

Published on: Sep 4, 2024

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

Keywords:

stochastic climate models,

El Niño Southern Oscillations,

Wiener chaos expansion,

Monte Carlo simulations

© 2024 Yusuf Aydogdu, N. Sri Namachchivaya, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 76 (2024): Issue 1

A Numerical Study of Stochastic El Niño Southern Oscillations Using Wiener Chaos Expansion and Monte Carlo Methods

Figures & Tables

Figure 1

Figure 2

Figure 3

Figure 4

Table 1

Figure 5

Figure 6

Figure 7

Figure 8

Table 2

Table 3

Figure 9

Figure 10

Figure 11

Figure 12

Paradigm

My account