MeridionalAdvectionDiffusion

General solver of the 1D meridional advection-diffusion equation on the sphere:

\[\begin{split}\frac{\partial}{\partial t} \psi(\phi,t) &= -\frac{1}{a \cos\phi} \frac{\partial}{\partial \phi} \left[ \cos\phi ~ F(\phi,t) \right] \\ F &= U(\phi) \psi(\phi) -\frac{K(\phi)}{a} ~ \frac{\partial \psi}{\partial \phi}\end{split}\]

for a state variable \(\psi(\phi,t)\), arbitrary diffusivity \(K(\phi)\) in units of \(x^2 ~ t^{-1}\), and advecting velocity \(U(\phi)\). \(\phi\) is latitude and \(a\) is the Earth’s radius (in meters).

\(K\) and \(U\) can be scalars, or optionally vector specified at grid cell boundaries (so their lengths must be exactly 1 greater than the length of \(\phi\)).

\(K\) and \(U\) can be modified by the user at any time (e.g., after each timestep, if they depend on other state variables).

A fully implicit timestep is used for computational efficiency. Thus the computed tendency \(\frac{\partial \psi}{\partial t}\) will depend on the timestep.

In addition to the tendency over the implicit timestep, the solver also calculates several diagnostics from the updated state:

diffusive_flux given by \(-\frac{K(\phi)}{a} ~ \frac{\partial \psi}{\partial \phi}\) in units of \([\psi]~[x]\)/s
advective_flux given by \(U(\phi) \psi(\phi)\) (same units)
total_flux, the sum of advective, diffusive and prescribed fluxes
flux_convergence (or instantanous scalar tendency) given by the right hand side of the first equation above, in units of \([\psi]\)/s

Non-uniform grid spacing is supported.

The state variable \(\psi\) may be multi-dimensional, but the diffusion will operate along the latitude dimension only.

class climlab.dynamics.meridional_advection_diffusion.MeridionalAdvectionDiffusion(K=0.0, U=0.0, use_banded_solver=False, prescribed_flux=0.0, **kwargs)[source]

Bases: AdvectionDiffusion

A parent class for meridional advection-diffusion processes.

Attributes:

K
U
current_time
depth: Depth at grid centers (m)
depth_bounds: Depth at grid interfaces (m)
diagnostics: Dictionary access to all diagnostic variables
elapsed_time
input: Dictionary access to all input variables
lat: Latitude of grid centers (degrees North)
lat_bounds: Latitude of grid interfaces (degrees North)
lev: Pressure levels at grid centers (hPa or mb)
lev_bounds: Pressure levels at grid interfaces (hPa or mb)
lon: Longitude of grid centers (degrees)
lon_bounds: Longitude of grid interfaces (degrees)
prescribed_flux
timestep: The amount of time over which step_forward() is integrating in unit seconds.
timestep_in_seconds: Return a float value representing the timestep in units of seconds

Methods

`add_diagnostic`(name[, value])	Create a new diagnostic variable called `name` for this process and initialize it with the given `value`.
`add_input`(name[, value])	Create a new input variable called `name` for this process and initialize it with the given `value`.
`add_subprocess`(name, proc[, verbose])	Adds a single subprocess to this process.
`add_subprocesses`(procdict)	Adds a dictionary of subproceses to this process.
`compute`()	Computes the tendencies for all state variables given current state and specified input.
`compute_diagnostics`([num_iter])	Compute all tendencies and diagnostics, but don't update model state.
`declare_diagnostics`(diaglist)	Add the variable names in `inputlist` to the list of diagnostics.
`declare_input`(inputlist)	Add the variable names in `inputlist` to the list of necessary inputs.
`integrate_converge`([crit, verbose])	Integrates the model until model states are converging.
`integrate_days`([days, verbose])	Integrates the model forward for a specified number of days.
`integrate_years`([years, verbose])	Integrates the model by a given number of years.
`remove_diagnostic`(name)	Removes a diagnostic from the `process.diagnostic` dictionary and also delete the associated process attribute.
`remove_subprocess`(name[, verbose])	Removes a single subprocess from this process.
`set_state`(name, value)	Sets the variable `name` to a new state `value`.
`step_forward`()	Updates state variables with computed tendencies.
`to_xarray`([diagnostics, timeave])	Convert process variables to `xarray.Dataset` format.

class climlab.dynamics.meridional_advection_diffusion.MeridionalDiffusion(K=0.0, use_banded_solver=False, prescribed_flux=0.0, **kwargs)[source]

Bases: MeridionalAdvectionDiffusion

A parent class for meridional diffusion-only processes, with advection set to zero.

Otherwise identical to the parent class.

Attributes:

K
U
current_time
depth: Depth at grid centers (m)
depth_bounds: Depth at grid interfaces (m)
diagnostics: Dictionary access to all diagnostic variables
elapsed_time
input: Dictionary access to all input variables
lat: Latitude of grid centers (degrees North)
lat_bounds: Latitude of grid interfaces (degrees North)
lev: Pressure levels at grid centers (hPa or mb)
lev_bounds: Pressure levels at grid interfaces (hPa or mb)
lon: Longitude of grid centers (degrees)
lon_bounds: Longitude of grid interfaces (degrees)
prescribed_flux
timestep: The amount of time over which step_forward() is integrating in unit seconds.
timestep_in_seconds: Return a float value representing the timestep in units of seconds

Methods

`add_diagnostic`(name[, value])	Create a new diagnostic variable called `name` for this process and initialize it with the given `value`.
`add_input`(name[, value])	Create a new input variable called `name` for this process and initialize it with the given `value`.
`add_subprocess`(name, proc[, verbose])	Adds a single subprocess to this process.
`add_subprocesses`(procdict)	Adds a dictionary of subproceses to this process.
`compute`()	Computes the tendencies for all state variables given current state and specified input.
`compute_diagnostics`([num_iter])	Compute all tendencies and diagnostics, but don't update model state.
`declare_diagnostics`(diaglist)	Add the variable names in `inputlist` to the list of diagnostics.
`declare_input`(inputlist)	Add the variable names in `inputlist` to the list of necessary inputs.
`integrate_converge`([crit, verbose])	Integrates the model until model states are converging.
`integrate_days`([days, verbose])	Integrates the model forward for a specified number of days.
`integrate_years`([years, verbose])	Integrates the model by a given number of years.
`remove_diagnostic`(name)	Removes a diagnostic from the `process.diagnostic` dictionary and also delete the associated process attribute.
`remove_subprocess`(name[, verbose])	Removes a single subprocess from this process.
`set_state`(name, value)	Sets the variable `name` to a new state `value`.
`step_forward`()	Updates state variables with computed tendencies.
`to_xarray`([diagnostics, timeave])	Convert process variables to `xarray.Dataset` format.