Held–Suarez forcing

Introduction

HeldSuarez implements the Held & Suarez (1994) benchmark forcing — the standard idealised test case for atmospheric dynamical cores. It replaces all physical parameterisations (radiation, convection, boundary layer) with two simple analytic terms:

Newtonian relaxation of temperature toward a prescribed radiative-equilibrium profile, and
Rayleigh friction that damps the winds in the boundary layer.

Coupled to a dynamical core, this forcing produces a realistic mid-latitude jet and eddy field without any expensive physics, which is exactly why it is the canonical benchmark for testing and intercomparing dynamical cores.

HeldSuarez is a TendencyComponent: it returns tendencies of temperature and the horizontal winds.

Physics

Thermal relaxation

Temperature is relaxed toward the equilibrium profile $T_{\text{eq}}$ at a latitude- and height-dependent rate $k_T$:

\[ \left(\frac{\partial T}{\partial t}ight)_{\text{HS}} = -k_T(\phi, \sigma)\,\big(T - T_{\text{eq}}(\phi, p)\big). \]

The equilibrium temperature combines a horizontal gradient (set by the equator–pole temperature difference $\Delta T_y$) and a vertical stratification (set by $\Delta\theta_z$), floored at 200 K:

\[ T_{\text{eq}} = \max\!\left[\,200,\; \left(315 - \Delta T_y\,\sin^2\phi - \Delta\theta_z\,\log\!\tfrac{p}{p_0}\,\cos^2\phiight)\left(\tfrac{p}{p_0}ight)^{\kappa}\,ight], \]

with $\kappa = R_d/c_p$. The relaxation rate is fast near the surface in the tropics and slow in the free atmosphere:

\[ k_T = k_a + (k_s - k_a)\,\max\!\left(0, \tfrac{\sigma - \sigma_b}{1 - \sigma_b}ight)\cos^4\phi, \]

where $\sigma = p/p_s$.

Rayleigh friction

The winds are damped within the boundary layer ($\sigma > \sigma_b$) at rate $k_v$:

\[ \left(\frac{\partial \mathbf{v}}{\partial t}ight)_{\text{HS}} = -k_v(\sigma)\,\mathbf{v}, \qquad k_v = k_f\,\max\!\left(0, \tfrac{\sigma - \sigma_b}{1 - \sigma_b}ight). \]

Above the boundary layer $k_v = 0$, leaving the free-atmosphere flow frictionless.

Constructor

climt.HeldSuarez(sigma_boundary_layer_top=0.7,
                 k_f=1/86400.0,
                 k_a=1/40.0/86400.0,
                 k_s=1/4.0/86400.0,
                 equator_pole_temperature_difference=60,
                 delta_theta_z=10)

Argument	Default	Symbol	Description
`sigma_boundary_layer_top`	`0.7`	$\sigma_b$	Top of the boundary layer in $\sigma$; friction and enhanced thermal relaxation act below it.
`k_f`	`1/86400` s⁻¹	$k_f$	Rayleigh friction rate (1 day⁻¹).
`k_a`	`1/(40{\cdot}86400)` s⁻¹	$k_a$	Free-atmosphere thermal relaxation rate (1/40 day⁻¹).
`k_s`	`1/(4{\cdot}86400)` s⁻¹	$k_s$	Surface tropical thermal relaxation rate (1/4 day⁻¹).
`equator_pole_temperature_difference`	`60`	$\Delta T_y$	Equilibrium equator–pole temperature contrast (K).
`delta_theta_z`	`10`	$\Delta\theta_z$	Equilibrium vertical potential-temperature contrast (K).

The reference pressure $p_0$, $c_p$ and $R_d$ come from climt’s registered constants (reference_air_pressure, etc.).

State

Role	Quantity	Dims	Units
in	`eastward_wind`, `northward_wind`	`[*, mid_levels]`	`m s^-1`
in	`air_temperature`	`[*, mid_levels]`	`degK`
in	`air_pressure`	`[*, mid_levels]`	`Pa`
in	`surface_air_pressure`	`[*]`	`Pa`
in	`latitude`	`[*]`	`degrees_north`
tendency	`eastward_wind`, `northward_wind`	`[*, mid_levels]`	`m s^-2`
tendency	`air_temperature`	`[*, mid_levels]`	`degK s^-1`

HeldSuarez has no diagnostics — it returns only tendencies.

Example

import climt
from climt import get_default_state, get_grid

forcing = climt.HeldSuarez()
state = get_default_state([forcing], grid_state=get_grid(nx=32, ny=16, nz=20))

tendencies, diagnostics = forcing(state)
print(tendencies["air_temperature"].values.shape)

In a full benchmark run, HeldSuarez is coupled to a dynamical core (the component supplies the physics tendencies; the dynamical core advances the state).

Source

Component: climt/_components/held_suarez.py

Reference

Held, I. M. & Suarez, M. J. (1994). A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteor. Soc. 75, 1825–1830.

--- title: "Held–Suarez forcing" --- ## Introduction `HeldSuarez` implements the **Held & Suarez (1994)** benchmark forcing — the standard idealised test case for atmospheric dynamical cores. It replaces all physical parameterisations (radiation, convection, boundary layer) with two simple analytic terms: 1. **Newtonian relaxation** of temperature toward a prescribed radiative-equilibrium profile, and 2. **Rayleigh friction** that damps the winds in the boundary layer. Coupled to a dynamical core, this forcing produces a realistic mid-latitude jet and eddy field without any expensive physics, which is exactly why it is the canonical benchmark for testing and intercomparing dynamical cores. `HeldSuarez` is a [`TendencyComponent`](component_types.qmd): it returns tendencies of temperature and the horizontal winds. ## Physics ### Thermal relaxation Temperature is relaxed toward the equilibrium profile $T_{\text{eq}}$ at a latitude- and height-dependent rate $k_T$: $$ \left(\frac{\partial T}{\partial t}ight)_{\text{HS}} = -k_T(\phi, \sigma)\,\big(T - T_{\text{eq}}(\phi, p)\big). $$ The equilibrium temperature combines a horizontal gradient (set by the equator–pole temperature difference $\Delta T_y$) and a vertical stratification (set by $\Delta\theta_z$), floored at 200 K: $$ T_{\text{eq}} = \max\!\left[\,200,\; \left(315 - \Delta T_y\,\sin^2\phi - \Delta\theta_z\,\log\!\tfrac{p}{p_0}\,\cos^2\phiight)\left(\tfrac{p}{p_0}ight)^{\kappa}\,ight], $$ with $\kappa = R_d/c_p$. The relaxation rate is fast near the surface in the tropics and slow in the free atmosphere: $$ k_T = k_a + (k_s - k_a)\,\max\!\left(0, \tfrac{\sigma - \sigma_b}{1 - \sigma_b}ight)\cos^4\phi, $$ where $\sigma = p/p_s$. ### Rayleigh friction The winds are damped within the boundary layer ($\sigma > \sigma_b$) at rate $k_v$: $$ \left(\frac{\partial \mathbf{v}}{\partial t}ight)_{\text{HS}} = -k_v(\sigma)\,\mathbf{v}, \qquad k_v = k_f\,\max\!\left(0, \tfrac{\sigma - \sigma_b}{1 - \sigma_b}ight). $$ Above the boundary layer $k_v = 0$, leaving the free-atmosphere flow frictionless. ## Constructor ``` python climt.HeldSuarez(sigma_boundary_layer_top=0.7, k_f=1/86400.0, k_a=1/40.0/86400.0, k_s=1/4.0/86400.0, equator_pole_temperature_difference=60, delta_theta_z=10) ``` | Argument | Default | Symbol | Description | |----------|---------|--------|-------------| | `sigma_boundary_layer_top` | `0.7` | $\sigma_b$ | Top of the boundary layer in $\sigma$; friction and enhanced thermal relaxation act below it. | | `k_f` | `1/86400` s⁻¹ | $k_f$ | Rayleigh friction rate (1 day⁻¹). | | `k_a` | `1/(40{\cdot}86400)` s⁻¹ | $k_a$ | Free-atmosphere thermal relaxation rate (1/40 day⁻¹). | | `k_s` | `1/(4{\cdot}86400)` s⁻¹ | $k_s$ | Surface tropical thermal relaxation rate (1/4 day⁻¹). | | `equator_pole_temperature_difference` | `60` | $\Delta T_y$ | Equilibrium equator–pole temperature contrast (K). | | `delta_theta_z` | `10` | $\Delta\theta_z$ | Equilibrium vertical potential-temperature contrast (K). | The reference pressure $p_0$, $c_p$ and $R_d$ come from `climt`'s registered constants (`reference_air_pressure`, etc.). ## State | Role | Quantity | Dims | Units | |------|----------|------|-------| | in | `eastward_wind`, `northward_wind` | `[*, mid_levels]` | `m s^-1` | | in | `air_temperature` | `[*, mid_levels]` | `degK` | | in | `air_pressure` | `[*, mid_levels]` | `Pa` | | in | `surface_air_pressure` | `[*]` | `Pa` | | in | `latitude` | `[*]` | `degrees_north` | | tendency | `eastward_wind`, `northward_wind` | `[*, mid_levels]` | `m s^-2` | | tendency | `air_temperature` | `[*, mid_levels]` | `degK s^-1` | `HeldSuarez` has no diagnostics — it returns only tendencies. ## Example ``` python import climt from climt import get_default_state, get_grid forcing = climt.HeldSuarez() state = get_default_state([forcing], grid_state=get_grid(nx=32, ny=16, nz=20)) tendencies, diagnostics = forcing(state) print(tendencies["air_temperature"].values.shape) ``` In a full benchmark run, `HeldSuarez` is coupled to a dynamical core (the component supplies the physics tendencies; the dynamical core advances the state). ## Source - Component: [`climt/_components/held_suarez.py`](https://github.com/CliMT/climt/blob/develop/climt/_components/held_suarez.py) ### Reference Held, I. M. & Suarez, M. J. (1994). *A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models.* Bull. Amer. Meteor. Soc. **75**, 1825–1830.