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An alpha-squared dynamo may use a tensorial alpha effect rather than an isotropic coefficient. For constant alpha tensor $\mathbf{\alpha }$ and magnetic diffusivity $\eta >0$, a steady nonzero Fourier mode with wavevector $\mathbf{K}$ satisfies $\eta K^2\mathbf{B}=i\mathbf{K}\times (\mathbf{\alpha }\mathbf{B})$ and $\mathbf{K}\cdot \mathbf{B}=0$. Anisotropy can change both the critical alpha magnitude and the preferred wavevector direction.

A Fourier mode has multiplier $e^{D{k}^{2}t}$, amplifying arbitrarily small high-frequency data. For a fixed positive time $t_0$, initial modes $e^{-D{n}^2{t}_0/2}\sin (nx)$ tend to zero in $L^2$ while their values at $t_0$ grow without bound. This violates continuous dependence on initial data and gives an ill-posed problem in ordinary unweighted spaces. An analytic-data restriction or physical short-scale regularization changes that conclusion.

Let fluid 1 lie above fluid 2, with mass densities ${\rho }_1<{\rho }_2$ and dynamic viscosities ${\mu }_1<{\mu }_2$. Define $\Delta \rho ={\rho }_2-{\rho }_1>0$, $\Delta \mu ={\mu }_2-{\mu }_1>0$, porosity $\varphi$, permeability of a porous medium $k$, and downward interface speed $W$. With Darcy flux $\varphi W$, no capillarity, and two unbounded layers, Darcy's law givesFor a Fourier mode of interface displacement, the two perturbation pressures decay exponentially into the respective layers. The kinematic boundary condition makes their amplitudes proportional to $\varphi {\mu }_1\sigma /(k|\alpha |)$ and $-\varphi {\mu }_2\sigma /(k|\alpha |)$. Linearized pressure continuity adds the base-gradient difference $\varphi W\Delta \mu /k-g\Delta \rho$. Eliminating the amplitudes proves the formula. The critical interface speed is $kg\Delta \rho /(\varphi \Delta \mu )$; the critical Darcy velocity is $kg\Delta \rho /\Delta \mu$. Without surface tension this model has unbounded short-wave growth rate on its unstable side.

Mixing by turbulence is represented approximately by a diffusive flux down a mean concentration gradient. Its coefficient is the eddy diffusivity. A constant-coefficient model gives a $D{\nabla }^2\sigma$ term and a smoothing rate $D{k}^2$ for a Fourier mode. Effective diffusion can suppress slow gravitational concentration even when microscopic diffusion is negligible.

In a thin Bottom Ekman layer, slowly evolving geostrophic flow drives upward Ekman pumping into a uniform-depth exterior layer. With $f>0$, $\alpha =\sqrt{f\nu /2}$ and $R_D=\sqrt{gH}/f$, its balanced height satisfies $({\nabla }_h^2-R_D^{-2}){\eta }_t=-(\alpha /H){\nabla }_h^2\eta$. Small-scale Fourier modes decay on $H/\alpha$; large-scale modes obey a diffusion equation with diffusivity $g\alpha /f^2$. The balanced approximation requires $|f|\tau \gg 1$.

For the two-field Keller--Segel model, put $a=D_2$, $b=D_{\rho }$, $C=D_1{f}_0\ge 0$, $c=\kappa -a_0{f}_0^{\prime }$. The upper growth rate is $s_{+}(z)=-[c+(a+b)z]/2+\sqrt{[c+(b-a)z{]}^2+4Cz}/2$. When $C>\max (ac,-bc)$ its unique maximum occurs at positive $z_{\ast }$ satisfying $s_{+}^{\prime }(z_{\ast })=0$. For equal diffusion coefficients $a=b$, this gives $z_{\ast }=(C^2/a^2-c^2)/(4C)$. For $c<0$ and $C\le -bc$, the fastest growth is homogeneous, with infinite wavelength. The finite-domain answer must maximize the growth rate over the permitted Fourier modes. Eliminating chemical dynamics instantaneously generally changes this selected scale.

For a circular domain of radius $R_0$ with line tension $\gamma$, the stiffness of a real polar sine or cosine amplitude with integer angular harmonic $m$ is $K_m=\pi \gamma (m^2-1)/R_0$, when the enclosed area is fixed. The equipartition theorem gives $⟨u_m^2⟩=k_{B}T{R}_0/[\pi \gamma (m^2-1)]$ for $m\ge 2$. A positive-frequency complex Fourier mode coefficient has half this variance. The uniform mode is forbidden and $m=1$ is translation.

The Fourier–Galerkin method seeks a finite Fourier series whose residual is orthogonal to every retained Fourier mode.

Orthogonality of integer-frequency Fourier modes gives this identity: the integral is one for $n=m$ and zero otherwise. It converts integer step-count constraints into products of generating functions, as in the dilute-hopping lattice propagator.

A Fourier spectral method represents a periodic unknown by finitely many Fourier modes and converts differentiation into multiplication by wavenumber.

For a translation-invariant difference operator $(Lu{)}_m={\sum }_j{a}_j{u}_{m+j}$, its Fourier symbol is $\ell (\theta )={\sum }_j{a}_j{e}^{ij\theta }$. Applying $L$ to a Fourier mode $e^{im\theta }$ multiplies that mode by $\ell (\theta )$. Symbols turn constant-coefficient stencil equations into scalar algebraic equations and expose their Von Neumann stability analysis.

For quadratic free energy with positive $a,\kappa$ and kinetic coefficient $\Gamma$, a deterministic initial Dirac delta distribution evolves under nonconserved order-parameter dynamics to the meanTaking the expectation removes the centered Gaussian white noise; the resulting diffusion-reaction equation is solved by the heat kernel. The integrated mean is $A{e}^{-\Gamma at}$, so spreading does not conserve the initial excess. The connected Fourier mode covariance grows from zero to $k_{B}T/(a+\kappa q^2)$ as $1-e^{-2\Gamma (a+\kappa q^2)t}$ by the Itô isometry. A realization retains equilibrium fluctuations after its mean has decayed. Pointwise continuum white-noise quantities require a coarse-graining cutoff.

After integrating a free-field momentum shell, restore the cutoff with $x^{\prime }=x/b$ and ${\varphi }^{\prime }(x^{\prime })=b^{(D-2)/2}{\varphi }_{<}(b{x}^{\prime })$. The gradient coefficient is invariant, the mass changes as $r^{\prime }=b^{2}r$, and a uniform conjugate field changes as $h^{\prime }=b^{(D+2)/2}h$. This follows by counting measure, derivative and field factors in the quadratic Hamiltonian. A nonuniform source instead obeys $h^{\prime }(x^{\prime })=b^{(D+2)/2}h(b{x}^{\prime })$, with the source on the right projected onto the retained Fourier modes. The field has engineering dimension $(D-2)/2$ and zero anomalous dimension.

An ill-posed problem fails existence, uniqueness or continuous dependence in the specified function spaces. Backward heat equation evolution is unstable because its Fourier modes grow like $e^{t|\xi {|}^2}$, making arbitrarily small high-frequency errors grow without a uniform bound.

For a homogeneous Keller--Segel model with nonnegative chemical production, linearized cell diffusion $D_2>0$, attraction coefficient $D_1>0$, and chemical diffusion $D_{\rho }>0$, put $c=\kappa -a_0{f}_0^{\prime }$ and $\kappa =(\rho k(\rho ){)}^{\prime }{|}_{{\rho }_0}$. A Fourier mode with squared wavenumber $z$ has determinant $z[D_2(c+D_{\rho }z)-D_1{f}_0]$. If $c\ge 0$, a growing mode on the infinite plane exists exactly when $D_1{f}_0>D_{2}c$, and $0<z<(D_1{f}_0-D_{2}c)/(D_2{D}_{\rho })$ is the unstable band. If $c<0$, the homogeneous chemical mode is already unstable. Dividing the threshold by $D_2\kappa$ to obtain a sum of two ratios is valid only when $\kappa >0$. Positivity of $k$ does not guarantee positivity of $(\rho k{)}^{\prime }$.

A linear fluctuating interface smooths height gradients by diffusion and receives additive Gaussian white noise. With periodic or Neumann boundary conditions, its nonzero Fourier modes are independent Ornstein-Uhlenbeck processes if the noise is diagonal in that basis. The unconstrained uniform height is a Brownian zero mode of a fluctuating interface.

Freezing coefficients locally in the Keplerian viscous diffusion equation gives ${\Sigma }_{1,t}\simeq 3{\partial }_{\Sigma }(\bar{\nu }\Sigma ){\Sigma }_{1,rr}$. A radial Fourier mode grows at rate $-3{\partial }_{\Sigma }(\bar{\nu }\Sigma )k_r^2$. Negative response reverses ordinary smoothing and gives a backward heat equation, within the wavelength range where the thin disc transport closure applies.

For a system or a multilevel method, each Fourier mode evolves through an amplification matrix $G(\theta )$. Stability requires bounds on $G(\theta {)}^n$ uniform in frequencies and mesh parameters. Eigenvalue moduli alone do not suffice: $G=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ has unit eigenvalues but $G^n$ has off-diagonal entry $n$. Uniformly bounded eigenvector matrices, or a uniformly positive energy symmetrizer, can supply the missing power bound.

For interaction $V=u\int {\varphi }^4/4!$, split $\varphi =\phi +\eta$ into slow and fast Fourier modes, with a centered fast Gaussian measure. The first cumulant expansion term is $⟨V⟩=(u/24)\int [{\phi }^4+6G_{>}(0){\phi }^2+3G_{>}(0{)}^2]$ by Wick theorem. The constant shifts the free energy, the quadratic term gives $\delta r=u{G}_{>}(0)/2$, and there is no gradient correction at this order. Rescaling gives $r^{\prime }=b^2[r+u{G}_{>}(0)/2]+O(u^2)$ and $u^{\prime }=b^{4-D}u+O(u^2)$. The mass shift mixes thermal and quartic perturbations; the interacting critical tuning need not have zero bare mass.

For centered fast Fourier modes and $V=u\int {\varphi }^4/4!$, the two-fast-field vertex is $(u/4)\int {\phi }^2{\eta }^2$. Its connected second cumulant uses $⟨{\eta }^2(x){\eta }^2(y){⟩}_c=2G_{>}(x-y{)}^2$, producing $-u^2\int {\phi }^2(x){\phi }^2(y)G_{>}(x-y{)}^2/16$. Extracting the local zero-external-momentum quartic coefficient gives $u^{\prime }=b^{4-D}(u-3u^2{I}_2/2)+\cdots$, where $I_2={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}K(p{)}^{-2}>0$. At $D=4$ its sign makes weak positive quartic coupling marginally irrelevant for increasing length scale, after tuning the critical mass. This local truncation omits generated derivative and higher-order operators; it is not the exact complete finite-shell effective action.