Asymptotic behavior of solutions to barotropic vorticity equation on a sphere

Let us denote by ℂ0∞(S)\mathbb{C}_{0}^{\infty}(S) the space of infinitely differentiable functions on the unit sphere S = {x ∈ ℝ³ : |x| = 1} which are orthogonal to any constant, and by (1)〈f,g〉=∫Sf(x)g(x)¯dS and ∥f∥=〈f,f〉1/2\langle f,g \rangle =\int_{S}f(x)\overline{g(x)}dS\ \ \ {\rm and}\ \ \ \| f \| =\langle f,f \rangle^{1/2} the inner product and the norm of functions of ℂ0∞(S)\mathbb{C}_{0}^{\infty}(S) , respectively. Denote by Ynm(λ,μ)Y_{n}^{m}\left(\lambda,\mu \right) the spherical harmonics (n ≥ 1, |m| ≤ n) that form the orthonormal system in ℂ0∞(S): 〈Ynm,Ylk〉=δmkδnl\mathbb{C}_{0}^{\infty}(S): \ \left\langle Y_{n}^{m},Y_{l}^{k}\right\rangle =\delta_{mk}\delta_{nl} where δ_mk is the Kronecker delta. Each spherical harmonic YnmY_{n}^{m} is the eigenfunction of the eigenvalue problem −ΔYnm=χnYnm , |m|≤n-\Delta Y_{n}^{m}=\chi_{n}{Y_{n}^{m}}, \ \ \left\vert m\right\vert \leq n for symmetric and positive definite spherical Laplace operator, where (2)χn=n(n+1)\chi_n = n (n+1)

Let us consider the non-stationary nonlinear BVE problem on S [11]: (4)Δψt+J(ψ,Δψ+2μ)+σΔψ−ν(−Δ)s+1ψ=F\Delta \psi_{t}+J\left(\psi,\Delta \psi +2\mu \right) +\sigma \Delta \psi-\nu (-\Delta)^{s+1}\psi =F(5)Δψ(0,x)=Δψ0(x) .\Delta \psi \left(0,x\right) =\Delta \psi_{0}\left(x\right)\ .

Here ψ is the stream function, Δψ is the relative vorticity, Δψ + 2μ is the absolute vorticity, F(t, x) is the forcing, σΔψ is the Rayleigh friction in the planetary boundary layer, J(ψ,h)=(n→×∇ψ)⋅∇hJ(\psi,h)=(\overrightarrow{n}\times \nabla \psi)\cdot \nabla h is the Jacobian determinant, n→\overrightarrow{n} is the outward unit normal vector to the surface of the sphere S, while the nonlinear term J(ψ, Δψ) and linear term J (ψ, 2μ) = 2ψ_λ represent the advection and the rotation of sphere, respectively. The velocity vector v→=n→×∇ψ\overrightarrow{v}=\overrightarrow{n}\times \nabla \psi is nondivergent: ∇⋅v→=0\nabla \cdot \overrightarrow{v}=0 . The viscosity is modeled by the term ν(−Δ)^s+1ψ, where −Δ is the spherical Laplace operator and s ≥ 1 is a real number.

Note that if ψ is a BVE solution then ψ + const is also the solution. We ignore this constant by searching a solution in spaces of functions orthogonal to a constant on the sphere. Spaces of functions in which a solution exists are important in many applications, and, in particular, in studying the stability of solutions.

In this section, we formulate two theorems proved in [11].

Here M is the constant from the estimate |J(ψ, h)| ≤ M ||Δψ|| ||Δh|| (see [11]). The case s = 1 and σ = 0 was proved in [4, 5], whilst the cases s = 1 and s = 2 (σ ≠ 0) were proved in [10]. Theorem 2 considers the general case when s ≥ 1 is a real number.

Let us study the asymptotic behavior of the BVE solutions as t → ∞.

Q.E.D.

According to (12) and (17), if some solution ψ belongs to the set B at time t₀ then it will belong to B for all t > t₀. Hence, all steady and periodic solutions (if they exist) belong to the set B. Evidently, the set B contains the maximal BVE attractor [14]. Theorem 3 is also valid if F(t,x)∈C(0,ω;ℍ0r)F(t,x)\in C(0,\omega;\ \mathbb{H}_{0}^{r}) where ω is the period. In this case, one should only replace in (7) and (8) the norms ||F||_r by the norms maxt∈0,ω]‖F‖r\underset{t\in \lbrack 0,\omega ]}{\max} \left\Vert F\right\Vert_{r} .

We now introduce a positive functional for estimating arbitrary perturbations of the BVE solution. Let ψ˜(t,λ,μ)\widetilde{\psi}(t,\lambda,\mu) be a real BVE solution. Then a perturbation ψ′(t, λ, μ) of the solution ψ˜\widetilde{\psi} satisfies the equation (18)∂∂tΔψ'+J(ψ',Δψ˜)+J(ψ˜,Δψ')+2∂ψ'∂λ+J(ψ',Δψ')=−[σ+νΛ2s]Δψ'\frac{\partial}{\partial t}\Delta \psi^{\prime}+J\left(\psi^{\prime},\Delta \widetilde{\psi}\right) +J(\widetilde{\psi},\Delta \psi^{\prime})+2\frac{\partial \psi^{\prime}}{\partial \lambda}+J\left(\psi^{\prime},\Delta \psi^{\prime}\right)=-[\sigma +\nu \Lambda^{2s}]\Delta \psi^{\prime} where s ≥ 1, ν > 0 and σ ≥ 0. Taking the inner product (1) of equation (18) in series with ψ′ and Δψ′, then using (9), we obtain the integral equations (19)ddtK+〈J(ψ',Δψ'),ψ˜〉+2σK+ν‖Λs+1ψ'‖2=0\frac{d}{dt}K+\left\langle J(\psi^{\prime},\Delta \psi^{\prime}),\widetilde{\psi}\right\rangle +2\sigma K+\nu \left\Vert \Lambda^{s+1}\psi^{\prime}\right\Vert^{2}=0 and (20)ddtη−〈J(ψ',Δψ'),Δψ˜〉+2ση+ν‖Λs+2ψ'‖2=0\frac{d}{dt}\eta -\left\langle J(\psi^{\prime},\Delta \psi^{\prime}),\Delta \widetilde{\psi}\right\rangle +2\sigma \eta +\nu \left\Vert\Lambda^{s+2}\psi^{\prime}\right\Vert^{2}=0 for the kinetic energy K(t)=12‖∇ψ'(t,x)‖2=12‖Λψ'(t,x)‖2K(t)=\frac{1}{2}\left\Vert \nabla \psi^{\prime}(t,x)\right\Vert^{2}=\frac{1}{2}\left\Vert \Lambda \psi^{\prime}(t,x)\right\Vert^{2} and enstrophy η(t)=12‖Δψ'(t,x)‖2\eta (t)=\frac{1}{2}\left\Vert \Delta \psi^{\prime}(t,x)\right\Vert^{2} of perturbation ψ′, respectively.

One can see from (19) and (20) that the first Jacobian in (18) does not affect the behavior of the perturbation energy K(t), while the second Jacobian in (18) does not affect the perturbation enstrophy η(t). Moreover, the sphere rotation and nonlinear term (the last two terms in the LHS of (18)) do not affect the behavior of K(t) and η(t).

We will estimate the behavior of perturbations ψ′(t, λ, μ) of a basic flow ψ˜(t,λ,μ)\widetilde{\psi}(t,\lambda,\mu) using the functional Q(t)=pK(t)+qη(t)=12(p‖∇ψ'‖2+q‖Δψ'‖2)Q(t)=pK(t)+q\eta (t)=\frac{1}{2}(p\left\Vert \nabla \psi^{\prime}\right\Vert^{2}+q\left\Vert \Delta \psi^{\prime}\right\Vert^{2}) where p and q are non-negative real numbers, not equal to zero simultaneously. Multiplying (19) and (20) by p and q, respectively, and combining the results, we obtain (21)ddtQ(t)=−2σQ(t)−R(t)−νp‖Λs+1ψ'‖2−νq‖Λs+2ψ'‖2\frac{d}{dt}Q(t)=-2\sigma Q(t)-R(t)-\nu p\left\Vert \Lambda^{s+1}\psi^{\prime}\right\Vert^{2}-\nu q\left\Vert \Lambda^{s+2}\psi^{\prime}\right\Vert^{2} where (22)R(t)=〈J(ψ',Δψ'),pψ˜−qΔψ˜〉 .R(t)=\left\langle J(\psi^{\prime},\Delta \psi^{\prime}),\ p\widetilde{\psi}-q\Delta \widetilde{\psi}\right\rangle\ .

Applying Lemma 2 we get −‖Λs+1ψ'‖2≤−2s‖∇ψ'‖2, −‖Λs+2ψ'‖2≤−2s‖Δψ'‖2 .-\left\Vert \Lambda^{s+1}\psi^{\prime}\right\Vert^{2}\leq -2^{s}\left\Vert \nabla \psi^{\prime}\right\Vert^{2},\qquad -\left\Vert \Lambda^{s+2}\psi^{\prime}\right\Vert^{2}\leq -2^{s}\left\Vert \Delta \psi^{\prime}\right\Vert^{2}\ .

Therefore, (21) leads to (23)ddtQ(t)≤−2ρQ(t)−R(t)\frac{d}{dt}Q(t)\leq -2\rho Q(t)-R(t) where ρ is defined by (11).

Let us consider three examples when the basic solution is zero or represents meteorologically important flows, such as super-rotation, or a homogeneous spherical polynomial. Each basic solution is assumed to be supported by appropriate forcing.

Let us obtain sufficient conditions for the BVE solution to be a global attractor.

First, assume that the basic solution ψ˜(t,λ,μ)\widetilde{\psi}(t,\lambda,\mu) of equation (4) is rather smooth, such that (25)p=supt≥0max(λ,μ)∈S|∇Δψ˜(t,λ,μ)| and q=supt≥0max(λ,μ)∈S|∇ψ˜(t,λ,μ)|p=\underset{t\geq 0}{\sup}\underset{(\lambda,\mu)\in S}{\max}\left\vert\nabla \Delta \widetilde{\psi}(t,\lambda,\mu)\right\vert\ \ \ {\rm and} \ \ \ q=\underset{t\geq 0}{\sup}\underset{(\lambda,\mu)\in S}{\max} \left\vert \nabla \widetilde{\psi}(t,\lambda,\mu)\right\vert are bounded. Then using the inequality |J(ψ, h)| ≤ |∇ψ| · |∇h| we get (26)|R(t)|=|〈J(pψ˜−qΔψ˜,ψ'),Δψ'〉|≤2pq‖∇ψ'‖‖Δψ'‖≤2pqQ(t) .\left\vert R(t)\right\vert =\left\vert \left\langle J(p\widetilde{\psi}-q\Delta \widetilde{\psi},\psi^{\prime}),\Delta \psi^{\prime}\right\rangle \right\vert\leq 2pq\left\Vert \nabla \psi^{\prime}\right\Vert \left\Vert \Delta \psi^{\prime}\right\Vert \leq 2\sqrt{pq}Q(t)\ .

The use of inequality (26) in (23) leads to

Note that in a limited domain on the plane, the condition for the global asymptotic stability of a smooth BVE solution was earlier obtained in [13] under the condition that rotation and linear drag are not taken into account (σ = 0) and s = 1. Theorem 4 extends this result to smooth flows on a rotating sphere when s ≥ 1, and the linear drag is also taken into account (σ ≠ 0).

It should be emphasized that in both assertions, the basic solution ψ˜(t,λ,μ)\widetilde{\psi}(t,\lambda,\mu) must have continuous derivatives up to the third order. But Theorems 1 and 2 guarantee the existence of a solution that has continuous derivatives only up to the second order. Theorem 5 of this section gives new global asymptotic stability conditions for smooth solution of equation (4), in which the requirement on the smoothness is weakened. These instability conditions are in full accordance with Theorem 1.

We now show that the restriction (25) on the smoothness of the basic solution can be weakened to be consistent with the solvability theorem (Theorem 1). To this end, let us consider a smooth solution ψ˜(t,λ,μ)\widetilde{\psi}(t,\lambda,\mu) such that (28)p=suptmax(λ,μ)∈S|Δψ˜(t,λ,μ)| and q=suptmax(λ,μ)∈S|ψ˜(t,λ,μ)|p=\underset{t}{\sup}\underset{(\lambda,\mu)\in S}{\max}\left\vert \Delta \widetilde{\psi}(t,\lambda,\mu)\right\vert \ \ \ {\rm and} \ \ q=\underset{t}{\sup}\underset{(\lambda,\mu)\in S}{\max}\left\vert \widetilde{\psi}(t,\lambda,\mu)\right\vert are bounded. Using the ɛ-inequality one can estimate R(t) as: (29)|R(t)|≤2pq‖∇ψ'‖‖∇Δψ'‖=2pq‖∇ψ'‖‖Λ3ψ'‖≤2pq‖∇ψ'‖‖ψ'‖3=(pq‖ψ'‖1)(2pq‖ψ'‖3)≤2qɛ2Q(t)+pqɛ2‖ψ'‖32 .\matrix{\left\vert R(t)\right\vert \leq 2pq\left\Vert \nabla \psi^{\prime}\right\Vert \left\Vert \nabla \Delta \psi^{\prime}\right\Vert=2pq\left\Vert \nabla \psi^{\prime}\right\Vert \left\Vert \Lambda^{3}\psi^{\prime}\right\Vert \leq 2pq\left\Vert \nabla \psi^{\prime}\right\Vert\left\Vert \psi^{\prime}\right\Vert_{3} \\ \quad\quad\quad\quad\quad\ \ = \left(\sqrt{pq}\left\Vert \psi^{\prime}\right\Vert_{1}\right) \left(2\sqrt{pq}\left\Vert \psi^{\prime}\right\Vert_{3}\right) \leq2q\varepsilon^{2}Q(t)+\frac{pq}{\varepsilon^{2}}\left\Vert \psi^{\prime}\right\Vert_{3}^{2}\ .}

In addition, the use of Lemma 2 in (21) leads to (30)ddtQ(t)≤−2ρQ(t)−R(t)−νp‖ψ'‖s+12−νq‖ψ'‖s+22≤−2ρQ(t)−R(t)−νq‖ψ'‖s+22≤−2ρQ(t)−R(t)−νqa1−s‖ψ'‖32\matrix{\frac{d}{dt}Q(t)\leq -2\rho Q(t)-R(t)-\nu p\left\Vert \psi^{\prime}\right\Vert_{s+1}^{2}-\nu q\left\Vert \psi^{\prime}\right\Vert_{s+2}^{2} \leq -2\rho Q(t)-R(t)-\nu q\left\Vert \psi^{\prime}\right\Vert_{s+2}^{2} \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\leq -2\rho Q(t)-R(t)-\nu qa^{1-s}\left\Vert \psi^{\prime}\right\Vert_{3}^{2}\hfill} where a=2a=\sqrt{2} . Combining (29) with (30) and setting ɛ² = pa^s−1/ν in order to eliminate the two terms containing ‖ψ'‖32\left\Vert \psi^{\prime}\right\Vert_{3}^{2} , we prove the following assertion:

In contrast to Theorem 4, Theorem 5 requires a non-zero viscosity coefficient ν. According to conditions (28), the basic solution must have continuous derivatives only up to the second order. Therefore, (31) can be applied to a wider class of BVE solutions. For example, the main solution to problem (4)–(5) can be one of the non-stationary modons [7, 15] supported, despite the dissipation, by the corresponding external forcing. As it is known, ∇Δψ˜(t,λ,μ)\nabla \Delta \widetilde{\psi}(t,\lambda,\mu) is discontinuous at the boundary between the inner and outer regions of the modon. Therefore, condition (31) is applicable, while condition (27) is not.

It was shown in [3] that in the case of a stationary forcing, the Hausdorff dimension of the global attractor of the barotropic vorticity.equation (4) on a sphere for s = 1 and s = 2 is limited by the generalized Grashof number (33)G(s)=‖F(x)‖/χ12s−1ν2(s)G(s)=\left\Vert F(x)\right\Vert /\chi_{1}^{2s-1}\nu^{2}(s) where χ₁ = 2 is the smallest positive eigenvalue of the spherical Laplace operator (see (2)).

This is an important hydrodynamic result. However doubts arise about its applicability to the dynamics of large-scale barotropic processes in the atmosphere. Indeed, the BVE forcing represents the influence of small-scale baroclinic processes (convection, etc.) on the large-scale dynamics of the barotropic atmosphere, and therefore is essentially nonstationary with rather complex spatio-temporal behavior. In order to show that the dimension of the global BVE attractor crucially depends on the space-time structure of the forcing, we now estimate the dimension of the global BVE attractor provided that the forcing of the BVE model is a quasiperiodic function in time and a homogeneous spherical polynomial of degree n in space: (34)F(t,x)=∑m=−nnFm(t)Ynm(x)F(t,x)=\sum\limits_{m=-n}^{n}F_{m}(t)Y_{n}^{m}\left(x\right) where (35)Fm(t)=fmexp{imωmt}, |m|≤n,F_{m}(t)=f_{m}\exp \left\{im\omega_{m}t\right\},\ \ \ \ \left\vert m\right\vert \leq n, each f_m is a constant, and the frequencies ω_m are incommensurate. Thus, F(t, x) ∈ H_n. Note that (36)‖F(t,x)‖=(∑m=−nn| fm|2)1/2≡Const\left\Vert F(t,x)\right\Vert =\left(\sum\limits_{m=-n}^{n}\left\vert\,f_{m}\right\vert^{2}\right)^{1/2}\equiv Const i.e., the norm (36) of forcing and the generalized Grashof number (33) are time-independent constants. Obviously, there are many quasiperiodic forcings of the form (34) which have the same norm (36), or the same Grashof number (33), but differ in degrees n or/and amplitudes f_m.

Since J(ψ, Δψ) = 0 for any function ψ ∈ H_n, there exists an exact BVE solution ψ˜(t,x)=∑m=−nnψ˜m(t)Ynm(x)\widetilde{\psi}(t,x)=\sum\limits_{m=-n}^{n}\widetilde{\psi}_{m}(t)Y_{n}^{m}\left(x\right) from the subspace H_n defined by the Fourier coefficients (37)ψ˜m(t)=AmFm(t)≡−{[σ+νχns]χn+im(χnωm−2)}−1Fm(t), |m|≤n .\widetilde{\psi}_{m}(t)=A_{m}F_{m}(t)\equiv -\left\{\left[ \sigma +\nu\chi_{n}^{s}\right] \chi_{n}+im(\chi_{n}\omega_{m}-2)\right\}^{-1}F_{m}(t), \ \ \ \ \left\vert m\right\vert \leq n\ .

Besides, it was shown in Example 3 that the subspace H_n is the domain of attraction of this solution.

Since the frequencies ω_m are rationally independent, the solution ψ˜(t,x)\widetilde{\psi}(t,x) is quasiperiodic, and its path is an open endless spiral densely wound around a 2n-dimensional torus in the (2n + 1)-dimensional complex space H_n. According to Theorem 3 in [9], the closure of this trajectory coincides with the torus. Hence, the Hausdorff dimension of the attractive set, that is solution (37), coincides with that of the torus and equals 2n.

Sufficient conditions for the global asymptotic stability of smooth solution (37) are given by Theorem 4. In our case ψ˜(t,x)∈Hn\widetilde{\psi}(t,x)\in \mathbf{H}_{n} , and (25) gives p = χ_nq = n(n + 1)q. Therefore, condition (27) for the global asymptotic stability of solution (37) accepts the form (38)σ+2sν>qχn\sigma +2^{s}\nu >q\sqrt{\chi_{n}} where (39)q=supt≥0max(λ,μ)∈S|∇ψ˜(t,λ,μ)| .q=\underset{t\geq 0}{\sup}\underset{(\lambda,\mu)\in S}{\max}\left\vert \nabla \widetilde{\psi}(t,\lambda,\mu)\right\vert\ .

Due to Lemma 1, (40)|∇ψ˜(t,x)|2=|∑m=−nnψ˜nm(t)∇Ynm(x)|2≤‖ψ˜(t,x)‖2{∑m=−nn|∇Ynm(x)|2}=χn Kn2 ‖ψ˜(t,x)‖2 .\left\vert \nabla \widetilde{\psi}(t,x)\right\vert^{2}=\left\vert\sum\limits_{m=-n}^{n}\widetilde{\psi}_{n}^{m}(t)\nabla Y_{n}^{m}(x)\right\vert^{2}\leq \left\Vert \widetilde{\psi}(t,x)\right\Vert^{2}\left\{\sum\limits_{m=-n}^{n}\left\vert \nabla Y_{n}^{m}(x)\right\vert^{2}\right\}=\chi_{n}\,K_{n}^{2}\,\left\Vert \widetilde{\psi}(t,x)\right\Vert^{2}\ .

Also, according to (37), (41)‖ψ˜(t,x)‖≤1(σ+νχns)χn‖F‖\left\Vert \widetilde{\psi}(t,x)\right\Vert \leq \frac{1}{(\sigma +\nu \chi_{n}^{s})\chi_{n}}\left\Vert F\right\Vert where the norm ||F|| of forcing (34), (35) is time-independent (see (36)). Due to (39)–(41), qχn≤Knσ+νχns‖F‖q\sqrt{\chi_{n}}\leq \frac{K_{n}}{\sigma +\nu \chi_{n}^{s}}\left\Vert F\right\Vert and hence condition (38) is satisfied if σ+2sν>Kn[σ+νχns]‖F‖\sigma +2^{s}\nu >\frac{K_{n}}{\left[ \sigma +\nu \chi_{n}^{s}\right]}\left\Vert F\right\Vert and also if 2sν>Knνχns‖F‖ .2^{s}\nu >\frac{K_{n}}{\nu \chi_{n}^{s}}\left\Vert F\right\Vert\ .

Using the definition (33) where χ₁ = 2, the last inequality can be written in terms of the generalized Grashof number: (42)22−sχnsπ2n+1>G(s) .2^{2-s}\chi_{n}^{s}\sqrt{\frac{\pi}{2n+1}}>G(s)\ .

Since χns12n+1=ns(n+1)s12n+1≥12n2s−1/2\chi_{n}^{s}\frac{1}{\sqrt{2n+1}}=n^{s}(n+1)^{s}\frac{1}{\sqrt{2}\sqrt{n+1}}\geq \frac{1}{\sqrt{2}}n^{2s-1/2} condition (42) holds if (43)23/2−sπn2s−1/2>G(s) .2^{3/2-s}\sqrt{\pi}n^{2s-1/2}>G(s)\ .

Thus, we prove the following assertion:

In particular, for s = 2 and s = 1, solution (37) is the global BVE attractor if (44)2−1/2πn7/2>G(2) and 21/2πn3/2>G(1),2^{-1/2}\sqrt{\pi}n^{7/2}>G(2) \ \ {\rm and} \ \ \ 2^{1/2}\sqrt{\pi}n^{3/2}>G(1), respectively. Note that s = 1 corresponds to the Navier-Stokes equations.

It follows from (44) that for a fixed finite value of the generalized Grashof number G, it is always possible to determine such an integer n(G) that the spiral solution generated by any quasiperiodic forcing (34)–(35) from subspace H_n with n ≥ n(G) is a global BVE attractor. For example, if we take G(2)=‖F(x)‖/χ13 ν2(2)=1500G(2)=\left\Vert F(x)\right\Vert /\chi_{1}^{3}\,\nu^{2}(2) =1500 for the barotropic atmosphere (this value was used in [3]), then condition (44) is satisfied for any forcing (34)–(35) whose degree n is equal to or greater than 8.

The result obtained is not unexpected. Indeed, for a fixed coefficient v(s), the number G(s) is fixed if the L₂-norm (36) of the forcing is a constant independent of n. Let the amplitudes |f_m| of oscillations of forcing be nonzero for all m. Then they must decrease as n grows, and for a sufficiently large number n (or for sufficiently small amplitudes |f_m|), the viscosity v(s) can become sufficient to satisfy condition (27) for the global asymptotic stability of the quasiperiodic solution (37).

Thus, unlike the case of stationary forcing when the Hausdorff dimension of the global BVE attractor is limited above by the generalized Grashof number G [3], in the case of the quasiperiodic forcing (34), the Hausdorff dimension 2n of the global spiral attractor (37) is not limited by the generalized Grashof number G and can become arbitrarily large as the degree n of the BVE forcing increases.

This result is of particular meteorological interest, since it shows that the dimension of the global attractor in the barotropic atmosphere can be unlimited, even if the generalized Grashof number (33) is bounded. Thus, the dimension of the global attractor crucially depends not only on the generalized Grashof number, but also on the time-space structure of the BVE forcing. This also shows that the search for a global attractor of small dimension in the barotropic atmosphere [1] is theoretically unjustified due to the fact that forcing usually has a very complex structure with a huge number of degrees of freedom.