The Casesk = sp = sq In this case and Q ∼ kq(k — p)(q — p) , (3.31) |gkpq |2 2 |gkpq |2 2x = —2 |Usp| [kq +(k — p)(q — p)] — 2 |Bsp| [k(k — p) + q(q — p)] . (3.32)√For k < p < q we obtain Q < 0 and thus x + x2 — Q ≥ 0, leading to exponentially growing solutionsindependent of Hc(p) and the ratio|Usp|/|Bsp|. In the present case both velocity and magneticfieldmodes have positive and negativecontributions to the sign of x. For both local (k ' p ' q) and nonlocal (k << p ' q) interactions Q = 0 and the sign of x determines whether unstable solutions occur. For the nonlocal case onlythe magnetic field term is positive, and x has the form |gkpq|2 2 2x ' 2 kq(|Bsp| — |Usp| ) . (3.33)leading to unstable solutions if |Bsp| > |Usp|, while for local interactions no instability occurs as the only term in x that does not vanish isp—|gkpq|2|Us |2kq < 0.For k < q < p, the possibility of exponentially growing solutions depends on the ratio |Usp|/|Bsp| and on the relative magnitudes of the wavenumbers k,p and q, as now Q > 0. Since the magnetic field term in x is now positive, instabilities occur for |Usp|/|Bsp| < 1. If |Usp|/|Bsp| > 1 it depends also on the cross-helicity whether instabilities occur. For maximal Hc(p) one obtains x2 — Q > 0, hence the perturbations cannot grow exponentially. If Hc(p) = 0 and |Usp|/|Bsp| is not too small, instabilities will occur, depending also on the shape of the triad (see appendix B.2 for further details). In general, the smaller |Usp|/|Bsp| the more unstable is the solution.For p < k < q we obtain x < 0 and Q > 0, furthermore x2 — Q > 0 independent of |Usp|/|Bsp| and Hc(p) (see appendix B.1), thus no linear instabilities occur. Nonlocal interactions (p << k ' q) do not lead to instabilities, since p px ' —|gkpq|2[k2 — kp](|Us |2 + |Bs |2) < 0 . (3.34)