= Hasselmann stochastic climate model
{c}
{title2=$C\dot T=F-\lambda T+\eta$}
The <Hasselmann stochastic climate model> represents slowly responding climate variables driven by fast weather fluctuations as well as external forcing. <Klaus Hasselmann>'s 1976 paper separates fast and slow variables, retains fluctuations lost by simple averaging, and explains why stabilizing <climate feedbacks> are needed to prevent indefinitely growing variability. Its general framework is broader than a single-temperature equation.<Stochastic climate models Part I. Theory by Klaus Hasselmann>[Hasselmann, Stochastic climate models. Part I. Theory, 1976, Section 2; Section 7, summary points 1–4]{ref}
Consider an original scalar specialization for a spatially averaged <temperature> anomaly $T$, with an effective <ocean mixed layer> <heat capacity> per unit <area> $C$:
$$
C\frac{dT}{dt}=F(t)-\lambda T+\eta(t),\qquad C>0,\quad\lambda>0.
$$
Here $F$ is imposed <radiative forcing>, $\lambda$ is the net restoring coefficient after the chosen <climate feedbacks>, and $\eta$ represents random net <heat flux> from unresolved weather. The anomaly <energy> per unit <area> is $CT$, so its tendency equals the net input. The weather term represents exchange with omitted degrees of freedom; it does not assert that an isolated ocean creates <energy>. $F$ and $\eta$ have units $\mathrm{W\,m^{-2}}$, and $\lambda$ has units $\mathrm{W\,m^{-2}\,K^{-1}}$.
First set $\eta=0$, start from $T(0)=0$, and switch to constant $F_0>0$. The <integrating factor> $e^{\lambda t/C}$ gives
$$
T(t)=\frac{F_0}{\lambda}(1-e^{-t/\tau}),\qquad \tau=\frac C\lambda.
$$
For $C=2.0\times10^8\,\mathrm{J\,m^{-2}\,K^{-1}}$, $\lambda=2.0\,\mathrm{W\,m^{-2}\,K^{-1}}$ and $F_0=4.0\,\mathrm{W\,m^{-2}}$, the <relaxation time> is $10^8\,\mathrm s$, about $3.17$ years using 365.25 days per year. At $t=\tau$, $T=2(1-e^{-1})\,\mathrm K\simeq1.264\,\mathrm K$, with eventual warming $2\,\mathrm K$. Doubling $C$ doubles the response time but leaves this equilibrium unchanged. This separates thermal inertia from <climate sensitivity>.
Now set $F=0$ and idealize the weather forcing as <Gaussian white noise> with
$$
\mathbb E\eta(t)=0,\qquad
\mathbb E[\eta(t)\eta(t')]=2D\delta(t-t'),\qquad D>0.
$$
The <Dirac delta function> has inverse-time units, so $D$ has units $(\mathrm{W\,m^{-2}})^2\mathrm s$. The rigorous <stochastic differential equation> is
$$
dT=-\frac{\lambda}{C}T\,dt+\frac{\sqrt{2D}}{C}\,dW_t,
$$
where $W_t$ is <Brownian motion>. This is an <Ornstein-Uhlenbeck process>. In stationary initialization its <variance> and <autocovariance function> are
$$
v_\infty=\frac{D}{C\lambda},\qquad R_T(u)=v_\infty e^{-|u|/\tau}.
$$
Define the two-sided angular-frequency <power spectral density> by $R_T(u)=(2\pi)^{-1}\int_{-\infty}^{\infty}S_T(\omega)e^{i\omega u}\,d\omega$. The <Fourier transform> of the exponential <autocovariance function>, or the squared response to a sinusoidal input, gives
$$
S_T(\omega)=\frac{2D}{\lambda^2+C^2\omega^2},\qquad
\frac{S_T(\omega)}{S_T(0)}=\frac{1}{1+(\omega\tau)^2}.
$$
Thus fast fluctuations are suppressed; the climate output is red over the frequency range above the corner $|\omega|=1/\tau$, with a low-frequency plateau. This is a <colored noise> spectrum without a periodic driver. The corner is $f_c=1/(2\pi\tau)$ in cycles per second, not $1/\tau$. The continuous white input is only an approximation on times longer than weather's <correlation time>.
\Image[/stochastic-climate-response-and-spectrum.png]
{title=Thermal inertia delays forced warming and filters random weather forcing}
{description=Original one-box model: the step-forced mean approaches two kelvins, with one-standard-deviation spread from a fixed initial temperature and chosen white-noise strength. The right panel shows the analytic normalized two-sided spectrum, not measured climate data.}
For the same $C,\lambda$ and $D=10^8\,(\mathrm{W\,m^{-2}})^2\mathrm s$, $v_\infty=0.25\,\mathrm{K^2}$ and the stationary <standard deviation> is $0.50\,\mathrm K$. From fixed $T(0)=0$, the <variance> grows as $v_\infty(1-e^{-2t/\tau})$, even when the forced <expected value> is predictable. Integrating $S_T/(2\pi)$ over all angular <frequencies> returns $D/(C\lambda)$, checking the normalization independently. If $\lambda=0$ the unforced <variance> instead grows as $2Dt/C^2$; if $\lambda<0$, deviations amplify. Neither case has the stationary spectrum above.
A red spectrum alone cannot establish <attribution of recent climate change>: internal variability, changing forcing and measurement effects must be distinguished. This scalar model supplies a calculable baseline for variability and persistence, not a universal spectrum for all climate timescales or a full attribution analysis.
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