Lorenz-84

Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for F = 8.

• Name: lorenz_84
• Field: physics/climate
• Formalism: ordinary differential equations
• References: Lorenz 1984; Datseris and Parlitz 2022

Factory

@register_model(
    field='physics/climate',
    tags=('chaos', 'ode', 'atmosphere', 'general circulation'),
    references=('Lorenz 1984', 'Datseris and Parlitz 2022'),
    defaults={'until': 100.0, 'dt': 0.05},
)
def lorenz_84(
    x: float = 1.0,
    y: float = 1.0,
    z: float = 1.0,
    *,
    a: float = 0.25,
    b: float = 4.0,
    F: float = 8.0,
    G: float = 1.0,
) -> System:
    """Return Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for `F = 8`.

    The jet `x` and the cosine and sine phases `y` and `z` of the eddies follow `dx/dt = -y^2 - z^2 - a x + a F`,
    `dy/dt = x y - b x z - y + G`, and `dz/dt = b x y + x z - z`.

    Args:
        x:
            The initial strength of the westerly jet.

        y:
            The initial cosine phase of the eddies.

        z:
            The initial sine phase of the eddies.

        a:
            The damping of the jet relative to the eddies.

        b:
            The displacement of the eddies by the jet.

        F:
            The forcing of the jet by the equator-to-pole temperature contrast.

        G:
            The forcing of the eddies by the land-sea contrast.

    Returns:
        system:
            The model, with the states `atmosphere.x`, `atmosphere.y`, and `atmosphere.z`.

    Examples:
        >>> from skcomplex.models.physics.climate import lorenz_84
        >>> lorenz_84(G=1.25).elements[0].G
        1.25
    """
    return System(elements=[Lorenz84Element('atmosphere', x=x, y=y, z=z, a=a, b=b, F=F, G=G)])

Components

class Lorenz84Element(Element):
    """Atmosphere of Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave of eddies.

    The jet `x` and the cosine and sine phases `y` and `z` of the eddies follow `dx/dt = -y^2 - z^2 - a x + a F`,
    `dy/dt = x y - b x z - y + G`, and `dz/dt = b x y + x z - z`.

    Args:
        label:
            The label of the atmosphere.

        visual:
            How the element is drawn, or `None` for `default_visual_`.

        x:
            The initial strength of the westerly jet.

        y:
            The initial cosine phase of the eddies.

        z:
            The initial sine phase of the eddies.

        a:
            The damping of the jet relative to the eddies.

        b:
            The displacement of the eddies by the jet.

        F:
            The forcing of the jet by the equator-to-pole temperature contrast.

        G:
            The forcing of the eddies by the land-sea contrast.

    Examples:
        >>> from skcomplex.models.physics.climate import Lorenz84Element
        >>> Lorenz84Element('atmosphere').F
        8.0
    """

    def __init__(
        self,
        label: str,
        visual: VisualSpec | None = None,
        *,
        x: Annotated[float, StateSpec()] = 1.0,
        y: Annotated[float, StateSpec()] = 1.0,
        z: Annotated[float, StateSpec()] = 1.0,
        a: Annotated[float, ParameterSpec()] = 0.25,
        b: Annotated[float, ParameterSpec()] = 4.0,
        F: Annotated[float, ParameterSpec()] = 8.0,
        G: Annotated[float, ParameterSpec()] = 1.0,
    ) -> None:
        super().__init__(label, visual=visual)
        self.x = x
        self.y = y
        self.z = z
        self.a = a
        self.b = b
        self.F = F
        self.G = G

    def _get_laws(self, s: SimpleNamespace) -> list[BaseLaw]:
        """Return the forcing and damping of the jet and its exchange with the eddies.

        Args:
            s:
                The namespace of the symbols the laws can use.

        Returns:
            laws:
                The rate laws of the jet and of both phases of the eddies.
        """
        return [
            RateLaw(s.x, -(s.y**2) - s.z**2 - s.a * s.x + s.a * s.F),
            RateLaw(s.y, s.x * s.y - s.b * s.x * s.z - s.y + s.G),
            RateLaw(s.z, s.b * s.x * s.y + s.x * s.z - s.z),
        ]

Equations

atmosphere.x: diff(x_atmosphere(t), t) == F_atmosphere*a_atmosphere - a_atmosphere*x_atmosphere(t) - y_atmosphere(t)^2 - z_atmosphere(t)^2
atmosphere.y: diff(y_atmosphere(t), t) == -b_atmosphere*x_atmosphere(t)*z_atmosphere(t) + x_atmosphere(t)*y_atmosphere(t) + G_atmosphere - y_atmosphere(t)
atmosphere.z: diff(z_atmosphere(t), t) == b_atmosphere*x_atmosphere(t)*y_atmosphere(t) + x_atmosphere(t)*z_atmosphere(t) - z_atmosphere(t)