The underlying equations for fluid-dynamic models are hyperbolic partial differential equations. They can generally be written in the form $$ \frac{\partial}{\partial t} u(t) = D(u(t)) $$ where $D$ in some way evaluates the current state of the system and its spatial derivatives.

A numerical simulation then integrates this differential equation, to extrapolate from a start state $u(t_0) = u_0$ the time-dependent $u(t)|_{t>t_0}$.

Well, if we can do that, then surely we could also solve in the inverse time direction, by considering the equation $$ \frac{\partial}{\partial t} u(-t) = -D(u(t)) $$ and running the integrator with $\tilde t = -t$, $\tilde D = -D$?

Actually, you quickly run into problems when you try that. The operator $D$ can be characterised by its Jacobian, which basically tells you how pertubations in the state influence the derivative. Specifically, the complex eigenvalues of the Jacobian can tell you whether a small deviation will a) amplify over time (positive real part), or b) decay (negative real part), or c) just oscillate (purely imaginary).

For physical systems the eigenvalues tend to be mostly c) or a): you get a lot of wave-like solutions which propagate / oscillate over the system, and tend to decay over time. b) however is more tricky: if you start with a small deviation from the start state, the system will over time deviate ever more and and more. Now, this kind of thing is by no means unheard of especially in meteorology; it's the essence of a chaotic system. Storms can emerge and grow stronger over time, but only by scooping up energy that's already stored in the system. At some point they'll stop.

OTOH, you always have a lot of consistently negative real-part eigenvalues. These correspond to dissipative effects: small-scale pertubations generally are smoothed out to zero by the physical effects, e.g. winds have friction, mixing of air of different temperature averages out the differences, etc.. If you now run the simulation backwards, you turn those negative real parts into positive real parts, and that means the system is suddenly massively chaotic on all length scales. Small pertubation arise out of numerical uncertainties, and grow over all bounds. You would not only end up with states different from the actual weather a week ago, but with states that are completely unlike anything the weather has ever been like – huge, erratic temperature fluctuations and small vortices with crazy wind speeds.

The underlying equations for fluid-dynamic models are hyperbolic partial differential equations. They can generally be written in the form $$ \frac{\partial}{\partial t} u(t) = D(u(t)) $$ where $D$ in some way evaluates the current state of the system and its spatial derivatives.

A numerical simulation then integrates this differential equation, to extrapolate from a start state $u(t_0) = u_0$ the time-dependent $u(t)|_{t>t_0}$.

Well, if we can do that, then surely we could also solve in the inverse time direction, by considering the equation $$ \frac{\partial}{\partial t} u(-t) = -D(u(t)) $$ and running the integrator with $\tilde t = -t$, $\tilde D = -D$?

Actually, you quickly run into problems when you try that. The operator $D$ can be characterised by its Jacobian, which basically tells you how pertubations in the state influence the derivative. Specifically, the complex eigenvalues of the Jacobian can tell you whether a small deviation will a) amplify over time (positive real part), or b) decay (negative real part), or c) just oscillate (purely imaginary).

For physical systems the eigenvalues tend to be mostly c) or b): you get a lot of wave-like solutions which propagate / oscillate over the system, and tend to decay over time. a) however is more tricky: if you start with a small deviation from the start state, the system will over time deviate ever more and and more. Now, this kind of thing is by no means unheard of especially in meteorology; it's the essence of a chaotic system. Storms can emerge and grow stronger over time, but only by scooping up energy that's already stored in the system. At some point they'll stop.

OTOH, you always have a lot of consistently negative real-part eigenvalues. These correspond to dissipative effects: small-scale pertubations generally are smoothed out to zero by the physical effects, e.g. winds have friction, mixing of air of different temperature averages out the differences, etc.. If you now run the simulation backwards, you turn those negative real parts into positive real parts, and that means the system is suddenly massively chaotic on all length scales. Small pertubation arise out of numerical uncertainties, and grow over all bounds. You would not only end up with states different from the actual weather a week ago, but with states that are completely unlike anything the weather has ever been like – huge, erratic temperature fluctuations and small vortices with crazy wind speeds.