RO Parameters#

Class

Parameter (equation)

Parameter (code)

Unit

Physical meaning

Deterministic Parameters

Linear parameters

\(R\)

R

month\(^{-1}\)

Bjerknes feedback

\(F_1\)

F1

K m\(^{-1}\) month\(^{-1}\)

Delayed oceanic feedback efficiency

\(F_2\)

F2

m K\(^{-1}\) month\(^{-1}\)

Recharge/discharge efficiency

\(\varepsilon\)

epsilon

month\(^{-1}\)

Basin adjustment

Nonlinear parameters

\(b_T\)

b_T

K\(^{-1}\) month\(^{-1}\)

Deterministic quadratic nonlinearities

\(c_T\)

c_T

K\(^{-2}\) month\(^{-1}\)

Deterministic cubic nonlinearities

\(d_T\)

d_T

m\(^{-1}\) month\(^{-1}\)

Deterministic quadratic nonlinearities

\(b_h\)

b_h

K\(^{-2}\) m month\(^{-1}\)

Deterministic quadratic nonlinearities

Noise parameters

\(\sigma_T\)

sigma_T

K month\(^{-\frac{1}{2}}\) (n_T = 1)K month\(^{-1}\) (n_T = 0)

Stochastic forcing amplitude

\(\sigma_h\)

sigma_h

m month\(^{-\frac{1}{2}}\) (n_h = 1)m month⁻¹ (n_h = 0)

Stochastic forcing amplitude

\(B\)

B

K⁻¹

State-dependence noise

\(m_T\)

m_T

month⁻¹

red noise decorrelation rate

\(m_h\)

m_h

month⁻¹

red noise decorrelation rate

Noise option parameters

\(n_T\)

n_T

no unit

noise option

\(n_h\)

n_h

no unit

noise option

\(n_g\)

n_g

no unit

noise option

Noise forcing

\(\xi_T\)

xi_T

no unit

red noise

\(\xi_h\)

xi_h

no unit

red noise

\(w_T\)

w_T

month\(^{-\frac{1}{2}}\)

white noise

\(w_h\)

w_h

month\(^{-\frac{1}{2}}\)

white noise

External forcing

\(E_T\)

E_T

K month⁻¹

extental forcing

\(E_h\)

E_h

m month⁻¹

extental forcing

Variable

\(T\)

T

K

ENSO SST anomaly

\(h\)

h

m

ENSO therocline anomaly

Linear and Nonlinear Parameters:

\(R\) (R), \(F_1\) (F1), \(F_2\) (F2), \(\varepsilon\) (epsilon), \(b_T\) (b_T), \(c_T\) (c_T), \(d_T\) (d_T), \(b_h\) (b_h)

  • 'parameter': [] → interpreted as zero (e.g., 'R': [] is the same as 'R': [0.0])

  • 'parameter': [value] → only the annual mean is used (e.g., 'R': [-0.05] means the annual mean value of –0.05 is used for R)

  • 'parameter': [mean, amplitude, phase (in radians)] → annual mean and annual seasonality are used (e.g., 'R': [-0.05, 0.05, np.pi] means R has an annual mean of –0.05 and a seasonal variation with amplitude 0.05 and phase shift \(\pi\))

  • 'parameter': [mean, amp₁, phase₁ (in radians), amp₂, phase₂ (in radians)] → annual mean, annual seasonality, and semi-annual seasonality are used (e.g., 'R': [-0.05, 0.05, np.pi, 0.01, np.pi/2] means R has an annual mean of –0.05, annual amplitude 0.05 with phase \(\pi\), and semi-annual amplitude 0.01 with phase \(\pi/2\))

  • Any other form → invalid (e.g., 'parameter': [mean, amp] will raise an error)

Noise Parameters:

\(\sigma_T\) (sigma_T), \(\sigma_h\) (sigma_h), \(B\) (B), \(m_T\) (m_T), \(m_h\) (m_h)

  • These follow the same format as the linear and nonlinear parameters, meaning that seasonality can be included for RO simulations.

  • However, RO parameter fitting does not support seasonality in these parameters. Therefore, it is strongly recommended to use annual means only for consistency during RO fitting.

Noise Option Parameters:

\(n_T\) (n_T), \(n_h\) (n_h), \(n_g\) (n_g)

  • n_T and n_h are flags that specify the noise type for the \(T\) and \(h\) equations, respectively:

    • 0: red noise

    • 1: white noise

  • n_g is a flag for the multiplicative noise type in the \(T\) equation:

    • 0: linear multiplicative noise \((1 + B \cdot T)\)

    • 1: Heaviside-based multiplicative noise \((1 + B \cdot H(T) \cdot T)\)

    • 2: no multiplicative noise (i.e., additive only)

  • 'parameter': [integer] or 'parameter': integer is a valid option; any other format will raise an error.

  • Mixed noise types (e.g., n_T = 0, n_h = 1) are allowed in the solver but not allowed during parameter fitting. Therefore, it is strongly recommended to use a consistent noise type (either white or red) for both the \(T\) and \(h\) equations.